1) Trigonometric ratios for similar triangles state that the ratios of corresponding sides of similar triangles are equal. 2) The sine and cosine of complementary angles have the property of being not only equal but also complementary to one another. 3) trigonometric ratios can be used to solve for a missing side or angle of a right triangle.
What are trigonometric ratios?1. Trigonometric ratios for similar triangles state that the ratios of corresponding sides of similar triangles are equal. Therefore, the ratios of the sine, cosine, and tangent of the corresponding angles in similar triangles will also be equal.
2. The sine and cosine of complementary angles have the property of being not only equal but also complementary to one another. This means that the sine of an angle is equal to its complement's cosine, and vice versa.
3. Using the appropriate formula based on the given information, trigonometric ratios can be used to solve for a missing side or angle of a right triangle.
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Ram Purchased a flat at ₹1. 1 lakh and Prem purchased aplot of land worth ₹ 1. 1 lakh. The respective annual rates at which the prices of the flat and the plot increases were 10% and 5%. After two years they exchanged their belongings and one paid the other the difference. Then who paid to whom by how much?
Prem purchased a plot for ₹1.1 lakh and its price increased at an annual rate of 5%.Ram's flat had a higher value than Prem's plot, Prem paid Ram the difference, which was ₹0.118 lakh or ₹11,800. Ram's flat is more valuable than Prem's plot of land
After two years, the value of Ram's flat would be ₹1.1 lakh + (10% of ₹1.1 lakh × 2 years) = ₹1.43 lakh.
Similarly, the value of Prem's plot of land would be ₹1.1 lakh + (5% of ₹1.1 lakh × 2 years) = ₹1.21 lakh.
Therefore, the difference in value between Ram's flat and Prem's plot of land is ₹1.43 lakh - ₹1.21 lakh = ₹22,000.
Ram would have to pay ₹22,000 to Prem as Ram's flat is more valuable than Prem's plot of land.
Ram purchased a flat for ₹1.1 lakh and its price increased at an annual rate of 10%. After two years, the flat's value would be:
1.1 lakh * (1 + 0.1)^2 = 1.1 lakh * 1.21 = ₹1.331 lakh
Prem purchased a plot for ₹1.1 lakh and its price increased at an annual rate of 5%. After two years, the plot's value would be:
1.1 lakh * (1 + 0.05)^2 = 1.1 lakh * 1.1025 = ₹1.213 lakh
After exchanging their properties, the difference in value is:
₹1.331 lakh (flat) - ₹1.213 lakh (plot) = ₹0.118 lakh
Since Ram's flat had a higher value than Prem's plot, Prem paid Ram the difference, which was ₹0.118 lakh or ₹11,800.
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Charlie's teacher claims that he does not study and just guesses on exams. on an exam with 201 true-false questions, Charlie answered 53.7% of the questions correctly. Calculations using these results show that if he were really just guessing, there would be roughly one chance and seven that he would do this well. Is there a statistically significant evidence against the teachers claim that Charlie is just guessing? Why or why not?
Answer:
Step-by-step explanation:
True-False questions consists of 2 possible answer, which are 50% each, so basically is Charlie was just guessing 201 questions, which means his possibility of getting everything correct is [tex]\frac{1}{2^2^0^1}[/tex], which is really small, his chance of getting 53,7% correct is [tex]\frac{1}{2^2^0^1 * 0.537}[/tex] which is roughly arround [tex]\frac{1}{2^2^0^0}[/tex]
a growing farming conglomerate increases its water usage at a rate of 7% every year. if it used 28,390 megaliters of water this year, then how much water will it use 5 years from now?
The farming conglomerate will use approximately 36,912.51 megaliters of water 5 years from now.
To calculate the amount of water the farming conglomerate will use 5 years from now, we can use the given information that its water usage increases at a rate of 7% every year.
Let's denote the current water usage as W₀ and the water usage after 5 years as W₅.
We know that W₅ = W₀ * (1 + r)^n, where r is the growth rate (in decimal form) and n is the number of years.
In this case, W₀ = 28,390 megaliters, r = 7% = 0.07, and n = 5.
Substituting these values into the formula, we have:
W₅ = 28,390 * (1 + 0.07)^5
Calculating this expression, we get:
W₅ ≈ 28,390 * (1.07)^5 ≈ 36,912.51 megaliters
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In a regular polygon (all sides and angles are equal-sized), the measure of each exterior angle is 12.5%
of the measure of each Interior angle.
Write and solve an equation to find m, the measure of each exterior angle.
Equation
Answer
Using the given information, the measure of each exterior angle in the polygon is 20°
Measure of angles in a polygon
From the question, we are to determine the measure of each exterior angle in the polygon
Let the number of sides in the polygon be n.
Sum of exterior angles in a polygon = 360°
Thus,
Measure of one exterior angle = 360°/n
From the given information,
The measure of each exterior angle is 12.5% of the measure of each interior angle
The measure of each interior angle = (n - 2)180° / n
Thus,
360°/n = 12.5% of (n - 2)180° / n
360° = 12.5% of (n - 2)180°
Solve for n
360° = 0.125 × (n - 2)180°
360° = (n - 2)22.5°
360° = 22.5n° - 45°
22.5n° = 360° + 45°
22.5n° = 405°
n = 405°/22.5°
n = 18
Then,
Measure of each exterior angle = 360° / 18
Measure of each exterior angle = 20°
Hence, the measure of each exterior angle is 20°
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For the system shown, what is the value of x + 2y?
x + 1/3y = 0
6x - 2y = 6
Answer:
Step-by-step explanation:
x + [tex]\frac{y}{3}[/tex] = 0 ------ ( 1 )
6x - 2y = 6 ------ ( 2 )
6x + 2y = 0 ----- ( 3 )
6x - 2y = 6 ------ ( 4 )
Adding ( 3 ) and ( 4 ) ; we get ,
12x = 6
x = 1/2
putting value of x in (3) , we get ,
3 + 2y = 0
2y = -3
x + 2y = (1/2) - 3
= - 5/2
The temperatures (in degrees Fahrenheit) in Long Island recorded by the weather bureau over a week were 42, 49, 53, 55, 50, 47, and 52. Which measure should the weather bureau use to calculate how far apart the upper and lower quartiles of the week's daily temperatures were?
A
mode
B.
mean absolute deviation
C.
interquartile range
D.
mean
1. What is the link between the normal distribution and (Variation 2 (and egg roulette)), where we relied on statements such as "about two-thirds of observations are within 1 standard deviation of the mean"?
2. Do natural phenomena such as hemoglobin levels or the weight of ants really follow a normal distribution?
3. If you add up a large number of random events, you get a normal distribution. How large a number makes a normal distribution? 4. Are the results of studies due to adding up a large number of random events?
The normal distribution is often used to model real-world phenomena, and one of its important properties is that about two-thirds of observations fall within one standard deviation of the mean. This makes it useful for analyzing data sets with a large number of observations, such as in Variation 2 or egg roulette.
Many natural phenomena do not follow a perfectly normal distribution, but certain aspects of these phenomena may be modeled using a normal distribution with some degree of accuracy. For example, the distribution of hemoglobin levels in a population may not be perfectly normal, but it may be close enough to a normal distribution for practical purposes.
The central limit theorem states that as the sample size increases, the distribution of the sample means approaches a normal distribution, regardless of the distribution of the original population. The sample size required to see this effect depends on the shape of the original distribution, but generally, a sample size of 30 or more is often considered sufficient to see a normal distribution emerge.
The results of many studies may involve adding up a large number of random events, but it is important to note that this does not necessarily mean that the results are normally distributed. However, the central limit theorem suggests that the distribution of the means of many independent random variables will approach a normal distribution, which can be useful in statistical analysis.
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A bird feeder is in the shape of a cylinder. It has a volume of about 100
100
cubic inches. It has a radius of 2
2
inches. What is the approximate height of the bird feeder? Use 3. 14
3. 14
for pi
To find the height of the bird feeder, we can use the formula for the volume of a cylinder, which is V = πr^2h, where V is the volume, r is the radius, and h is the height of the cylinder. In this case, the volume is given as 100 cubic inches, and the radius is 2 inches.
Substituting the known values into the formula, we have:
100 = 3.14 × 2^2 × h
Simplifying the equation:
100 = 3.14 × 4 × h
100 = 12.56h
To solve for h, we divide both sides of the equation by 12.56:
h = 100 / 12.56
h ≈ 7.97
Therefore, the approximate height of the bird feeder is approximately 7.97 inches.
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I honestly have no clue what im saying here-
Answer:
the answer is lateral area
Sylvia is going to pick out a souvenir hat while she is on vacation. She knows the price of the hat will increase by 5% when she pays sales tax. She writes the expression h+0. 05h to find the total amount she will pay for the hat if it costs h dollars. Jin correctly writes another expression 1. 05h 5hat will also find the total amount Slavia will pay for the hat if it costs h dollars
The expression 1.05h correctly represents the total amount Sylvia will pay for the hat if it costs h dollars.
Sylvia knows that the price of the hat will increase by 5% when she pays sales tax. Let's break down the expressions to understand them better.
Expression 1: h + 0.05h
In this expression, h represents the cost of the hat. Adding 0.05h to h accounts for a 5% increase. However, this expression does not accurately represent the total amount Sylvia will pay because it only considers the increase due to sales tax, not the original cost of the hat.
Expression 2: 1.05h
In this expression, 1.05 represents the total cost multiplier, which includes the original cost of the hat and the 5% increase. Multiplying h by 1.05 accounts for both the original cost and the sales tax increase. This expression correctly calculates the total amount Sylvia will pay for the hat.
Therefore, the expression 1.05h is the correct representation of the total amount Sylvia will pay for the hat if it costs h dollars.
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During a sale, a store offered a 20% discount on a tablet computer that originally sold for $710. After the sale, the discounted price of the tablet computer was marked up by 20%. To the nearest whole number, what percent of the original price was the price after markup?
Answer:
Justin is packing a container with books the dimension of each book are 8 in by 6 in by 2 in the dimensions of the container are 16 in by 12 in by 12 in all of the books in the container are rectangular prisms how many books can fit in the container if the books are packed so that there is no unused space in the container
In the diagram below find m<1
Answer: 50°
Step-by-step explanation:
7 and 6 are vertical angles, which means they are equal. m<7 and m<6 are 130 degrees. Then. m<4 and m<6 are supplementary angles, and if m<6 is 130°, then you subtract that from 180 to get 50 = m<4. m<4 and m<1 are vertical angles, so that means m<1 is 50°.
given f(x)=3x+2 and g(x)= √x-1, determine the following: (f•g)(5) =
The value of (f•g)(5) is 21.
Given are two functions f(x) = 3x+2 and g(x)= √x-1, we need to determine (f•g)(5),
So, firstly determine (f•g)(x),
Therefore,
(f•g)(x) = (3x+2)(√x-1)
Now for, (f•g)(5) put x = 5,
We get,
(f•g)(5) = (3·5+2)(√5-1)
(f•g)(5) = 17 × 1.23
(f•g)(5) = 21
Hence the value of (f•g)(5) is 21.
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Choose the INCORRECT statement below.
Which question was answered with variable data?
Math item stem image
CLEAR CHECK
Answer:
Question 1
only, because there are many cards in a standard deck of playing cards.
Question 2
only, because the snowfall was not the same for every month.
Neither question, because in each data set, there is only one outcome.
Both questions, because each one has more than one answer.
The term "Variable data" refers to data that can vary or change. It typically includes numerical data that can take on different values or measurements, such as temperature, height, weight, or time.the correct answer is "Neither question, because in each data set, there is only one outcome."
Neither question was answered with variable data, because in each data set, there is only one outcome.
The term "variable data" refers to data that can vary or change. It typically includes numerical data that can take on different values or measurements, such as temperature, height, weight, or time.
In this case, neither question involves variable data. Question 1 is about the number of cards in a standard deck of playing cards, which is a fixed quantity and does not vary. Question 2 is about the total snowfall for each month, but each month has a single, fixed value, so there is no variation in the data.
Therefore, the correct answer is "Neither question, because in each data set, there is only one outcome."
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a 5 foot ladder leaning against a wall is slipping down the wall at 1 foot per minute. how fast is the base of the ladder moving away from the wall when the top of the ladder is 4 feet from the ground?
So the base of the ladder is moving away from the wall at a rate of 3/4 feet per minute when the top of the ladder is 4 feet from the ground.
We can use the Pythagorean Theorem to relate the ladder's length to the distance it is from the wall. Let's call the distance from the wall "x", and the length of the ladder "y". Then:
x^2 + y^2 = 5^2
We can take the derivative with respect to time of both sides of this equation, using implicit differentiation:
2x(dx/dt) + 2y(dy/dt) = 0
Now we can plug in the values we know and solve for the rate at which the base of the ladder is moving away from the wall (which is the same as the rate at which x is increasing):
x = sqrt(5^2 - 4^2) = 3
y = 4
dx/dt = 1 (given)
dy/dt = ?
Plugging in these values, we get:
2(3)(1) + 2(4)(dy/dt) = 0
Simplifying:
6 + 8(dy/dt) = 0
Solving for dy/dt:
dy/dt = -6/8 = -3/4
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Baka borrows $2,000 to buy a piano. The simple interest owed at the end of 3 years is $1,080. What is the annual interest rate on Baka’s loan for the piano?
The quadratic equation h=-16t^2+32t+2
represents the height, h (in feet), of a ball kicked after t seconds. Answer each question. Express each answer as a decimal rounded to the nearest hundredth.
How long will it take the ball to reach 18 feet?
When will the object be at 10 feet?
When will the ball hit the ground?
By solving the quadratic equations,
It will take the ball 1 second to reach 18 feet
It will take the ball 1.71 seconds to be at 10 feet
It will take the ball 2.06 seconds to hit the ground
Solving quadratic equations: Determining how long it would take the ball to reach a heightFrom the question, we are to determine how long it would take the ball to reach 18 feet
To determine how long it would take the ball to reach 18 feet, we will set h = 18 in the equation
The given equation is
h = -16t² + 32t + 2
Put h = 18
18 = -16t² + 32t + 2
Rearrange
16t² - 32t - 2 + 18 = 0
16t² - 32t + 16 = 0
16t² - 16t - 16t + 16 = 0
16t(t - 1) -16(t - 1) = 0
(16t - 16)(t - 1) = 0
16t - 16 = 0 OR t - 1 = 0
16t = 16 OR t = 1
t = 16/16 OR t = 1
t = 1 OR t = 1
Hence,
t = 1 second
Hence, it will take the ball 1 second to reach 18 feet
To determine how long it will take the ball to reach 10 feet, we will set h = 10
h = -16t² + 32t + 2
Put h = 10
10 = -16t² + 32t + 2
Rearrange
16t² - 32t - 2 + 10 = 0
16t² - 32t + 8 = 0
Using the quadratic formula,
t = 0.29 second OR t = 1.71 seconds
The ball will hit the ground when h = 0
Set h = 0 in the equation
h = -16t² + 32t + 2
0 = -16t² + 32t + 2
16t² - 32t - 2 = 0
Using the quadratic formula,
t = -0.06 OR t = 2.06
Since t cannot be negative,
t = 2.06 seconds
Hence, it will take 2.06 seconds for the ball to hit the ground
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pls asnwer the circled bit pls!
worth 35 points :))
Answer:
gradient = - 2 , y- intercept = 1
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope (gradient) and c the y- intercept )
given
2x + y = 1 ( subtract 2x from both sides )
y = - 2x + 1 ← in slope- intercept form
with gradient m = - 2 and y- intercept c = 1
helpp A car manufacturer claims that the average top speed of its newest cars is 145 mph. A sample of 87 cars showed that their average top speed was 150 mph, with a sample standard deviation of 19 mph. Test this hypothesis using a 1% significance level.
a.Reject the null hypothesis. There is not enough evidence to oppose the car manufacturer's claim.
b.Fail to reject the null hypothesis. There is not enough evidence to oppose the car manufacturer's claim.
c.Reject the null hypothesis. There is enough evidence to oppose the car manufacturer's claim.
d.Fail to reject the null hypothesis. There is enough evidence to oppose the car manufacturer's claim.
The correct answer is (c) Reject the null hypothesis. There is enough evidence to oppose the car manufacturer's claim.
How to find the null hypothesisThe null hypothesis (H0) is that the true average speed is 145 mph, and the alternative hypothesis (H1) is that the average speed is not 145 mph.
Using a z-test, we can calculate the z-score as [tex](150-145)/(19 \sqrt(87)),[/tex]which is approximately 3.67.
For a two-tailed test at the 1% significance level, the critical z-scores are -2.576 and 2.576.
Since the calculated z-score is beyond this range, we can reject the null hypothesis.
Therefore, there is enough evidence to oppose the car manufacturer's claim.
The correct answer is (c) Reject the null hypothesis. There is enough evidence to oppose the car manufacturer's claim.
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Elizabeth goes to a restaurant and the subtotal on the bill was
�
x dollars. A tax of 6% is applied to the bill. Elizabeth decides to leave a tip of 21% on the entire bill (including the tax). Write an expression in terms of
�
x that represents the total amount that Elizabeth paid
The total amount Elizabeth paid can be expressed as 1.27x, where x is the subtotal on the bill.
To calculate the total amount, first, we need to add the tax to the subtotal, which gives us 1.06x. Then, we need to add the tip to the entire amount, which is 21% of the sum of the subtotal and the tax. This can be expressed as 0.21(1.06x) = 0.22386x. Finally, we add the tip to the subtotal and the tax, which gives us 1.06x + 0.22386x = 1.28386x. This is the same as multiplying the subtotal by 1.27, which gives us the expression 1.27x. Therefore, the total amount Elizabeth paid is 1.27 times the subtotal on the bill.
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please help plsssssss
1. 2x = 14 --> divide both sides by 2 --> x = 14/2 --> x = 7
2. x + 4 = 3 --> subtract 4 from both sides --> x = 3-4 --> x = -1
3. 3x + 6 = 12 --> subtract 6 from both sides --> 3x = 12-6 --> 3x = 6
4. 2x + 8 = 14 --> factor 2 out on the left side --> 2(x+4) = 14 --> divide both sides by 2 --> x + 4 = 14/2 --> x + 4 = 7
5. x + 5 = 4 --> factor -1 out on the left side --> -(-x-5) = 4 --> divide both sides by -1 --> -x-5 = 4/-1 --> -x-5 = -4
6. 2x + 4 = 3x - 5 --> subtract 2x from both sides --> 4 = 3x - 5 - 2x --> add 5 on both sides --> 4 + 5 = 3x - 2x --> 3x - 2x = 9
7. x - 8 = 24 --> add 8 on both sides --> x = 24 + 8 --> x = 32
8. 21 - 5x = 2 --> subtract 21 from both sides --> -5x = 2 - 21 --> -5x = -19 --> divide both sides by -1 --> 5x = -19/-1 --> 5x = 19
9. 4x + 2 = 8 --> add 2 on both sides --> 4x + 2 + 2 = 8 + 2 --> 4x + 4 = 10 --> factor 4 out on the left side --> 4(x+1) = 10 --> divide both sides by 4 --> x + 1 = 10 / 4 --> x + 1 = 2.5
10. (x-5) / 7 = 10 --> multiply both sides by 7 --> x-5 = 10 * 7 --> x-5 = 70
Solve the right triangle. If two sides are given, give angles in degrees and minutes.
A = 12° 33', c = 283 ft
Round side lengths to two decimal places.
A 12-sided solid has faces numbered 1 to 12. The table shows the results of rolling the solid 200 times. Find the experimental probability of rolling a number less than 3.
The experimental probability of rolling a number less than 3 is given as follows:
p = 3/200.
How to calculate a probability?A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.
The total number of outcomes for this problem is given as follows:
2000.
The desired outcomes are those with a result of 1 or 2, which are numbers less than 3, hence the amount is given as follows:
16 + 14 = 30.
Hence the experimental probability is given as follows:
p = 30/2000
p = 3/200.
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Y=3x+6 and 6x+2y=8. For what value x do the two equations each give the same value for y?
To find the value of x for which the two equations give the same value for y, we can set the expressions for y in both equations equal to each other and solve for x.
Given:
Equation 1: y = 3x + 6
Equation 2: 6x + 2y = 8
Substitute the expression for y from Equation 1 into Equation 2:
6x + 2(3x + 6) = 8
Simplify and solve for x:
6x + 6x + 12 = 8
12x + 12 = 8
12x = 8 - 12
12x = -4
x = -4/12
x = -1/3
Therefore, when x = -1/3, the two equations will have the same value for y.
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find the general solution of the given second-order differential equation. y'' − 6y' + 10y = 0
The general solution of the differential equation is: y = c1 e^(3t) cos(t) + c2 e^(3t) sin(t), where c1 and c2 are arbitrary constants determined by the initial or boundary conditions.
The given differential equation is a second-order homogeneous linear differential equation with constant coefficients. Its characteristic equation is obtained by assuming a solution of the form y = e^(rt), where r is a constant. Substituting this into the differential equation, we get:
r^2 e^(rt) - 6r e^(rt) + 10 e^(rt) = 0
Dividing both sides by e^(rt) (which is non-zero), we get:
r^2 - 6r + 10 = 0
Solving for r using the quadratic formula, we get:
r = (6 ± sqrt(6^2 - 4(1)(10))) / 2
r = 3 ± i
Therefore, the general solution of the differential equation is:
y = c1 e^(3t) cos(t) + c2 e^(3t) sin(t)
where c1 and c2 are arbitrary constants determined by the initial or boundary conditions.
This solution consists of a linear combination of two functions: the exponential function e^(3t) and the periodic function cos(t) or sin(t), which oscillates between -1 and 1. The exponential function represents the growth or decay of the system, while the periodic function represents the oscillations or vibrations of the system. The constants c1 and c2 determine the amplitude and phase of the oscillations, respectively. The behavior of the system depends on the signs and magnitudes of the real and imaginary parts of the roots, which determine whether the solutions are overdamped, critically damped, or underdamped.
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a teacher wants to create a graph of the students' scores on the final exam and see what the distribution looks like, looking for clusters and gaps in the scores. which type of graph is the most appropriate?
Answer:
A histogram would be the most appropriate graph to display the distribution of scores and identify clusters and gaps.
Step-by-step explanation:
a carpenter is building a rectangular room with a fixed perimeter of 600 feet. what are the dimensions of the largest room that can be built? what is its area? 150 ft by 150 ft; 22,500 ft2 300 ft by 300 ft; 90,000 ft2 60 ft by 540ft; 32,400 ft2 150 ft by 450 ft; 67,500 ft2
To find the dimensions of the largest room that can be built with a fixed perimeter of 600 feet.
We need to divide the perimeter by 2 and use that as the sum of two adjacent sides. Let's call the length of the rectangle "l" and the width "w".
So we have: 2l + 2w = 600
Simplifying: l + w = 300
We want to maximize the area of the rectangle, which is given by: A = lw
We can solve for one variable in terms of the other: l = 300 - w
Substituting into the area equation:
A = (300 - w)w
A = 300w - w^2
To maximize the area, we need to find the value of w that makes the derivative of A with respect to w equal to 0: dA/dw = 300 - 2w = 0
w = 150
So the width of the rectangle is 150 feet. Substituting back into the perimeter equation: l + 150 = 300
l = 150
So the length of the rectangle is also 150 feet.
Therefore, the largest room that can be built has dimensions 150 ft by 150 ft, and its area is: A = lw = 150 * 150 = 22,500 ft^2
The dimensions of the largest rectangular room a carpenter can build with a fixed perimeter of 600 feet are 150 ft by 150 ft. The area of this room is 22,500 ft². This is because when the length and width are equal, the area of the rectangle is maximized.
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use quantifiers and logical connectives to express the factthat every linear polynomial (that is, polynomial of degree 1) with real coefficients and where the coefficient ofx is nonzero, has exactly one real root.
The expression states that for every linear polynomial p with real coefficients and a nonzero coefficient of x, there is exactly one real root r.
For all linear polynomials with real coefficients and a nonzero coefficient of x, there exists exactly one real root. This can be expressed using the universal quantifier "for all" and the existential quantifier "there exists", connected by the logical connective "and". Additionally, the statement "exactly one real root" can be expressed using the quantifier "there exists" and the logical connective "and".
Using quantifiers and logical connectives, we can express the given fact as follows:
∀p ∃!r ((isLinearPolynomial(p) ∧ hasRealCoefficients(p) ∧ coefficientOfX(p) ≠ 0) → hasRealRoot(p, r))
Explanation:
- ∀p: For every polynomial p
- ∃!r: There exists exactly one real root r
- isLinearPolynomial(p): p is a linear polynomial (degree 1)
- hasRealCoefficients(p): p has real coefficients
- coefficientOfX(p) ≠ 0: The coefficient of x in p is nonzero
- hasRealRoot(p, r): p has a real root r
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Compute the zeroes of the polynomial 4x2 – 4x – 8. Also, establish a
relationship between the zeroes and coefficients.
The zero's of the polynomial 4x² - 4x - 8 are as follows:
x = 2 or x = -1
How to find zeros of a polynomial?Zeros of a polynomial can be defined as the points where the polynomial becomes zero as a whole. In other words, the zeros of a polynomial p(x) are all the x-values that make the polynomial equal to zero.
Therefore, let's find the zero's of the polynomial.
4x² - 4x - 8 = 0
divide through by 4
x² - x - 2 = 0
x² + x - 2x - 2 = 0
x(x + 1) -2(x + 1) = 0
(x - 2)(x + 1) = 0
Therefore,
x = 2 or x = -1
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The zeros of polynomial 4x² - 4x - 8, are 2 and -1. The relationship between the zeros and coefficients are 1 and -2.
Let the given polynomial be p(x) = 4x² - 4x - 8
To find zeros, take p(x) = 0
Factorizing the equation,
4x² - 4x - 8 = 0
4(x² - x - 4) = 0
x² - 2x + x- 2 = 0
x(x-2) + 1(x-2) = 0
(x-2) (x+1) = 0
∴ x=2 and x=-1
The roots of 4x² - 4x- 8 are 2 and -1
Relationship between the sum of zeros and coefficients:
-Coefficient of x/ coefficient of x²
(-1) + 2 = -(-4)/4 = 1
Relation between the product of zeros and coefficients:
Constant/ coefficient of x²
(-1) × 2 = -8/4 = -2
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