Answers
We have the following quadratic function:
[tex]f(x)=2x^2+7x-30[/tex]And we need to find its zeros i.e. the solutions to the equation:
[tex]2x^2+7x-30=0[/tex]Given a quadratic equation like the following:
[tex]ax^2+bx+c=0[/tex]Its solutions are given by the quadratic solving formula:
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]In our case we have a=2, b=7 and c=-30. Then we get:
[tex]x=\frac{-7\pm\sqrt[]{7^2-4\cdot2\cdot(-30)}}{2\cdot2}=\frac{-7\pm\sqrt[]{49+240}}{4}=\frac{-7\pm\sqrt[]{289}}{4}[/tex]So we continue:
[tex]x=\frac{-7\pm\sqrt[]{289}}{4}=\frac{-7\pm17}{4}[/tex]So we have two solutions:
[tex]\begin{gathered} x_1=\frac{-7+17}{4}=\frac{10}{4}=2.5 \\ x_2=\frac{-7-17}{4}=-\frac{24}{4}=-6 \end{gathered}[/tex]Then the answers are -6 and 2.5.
Related Questions
Donovan walked a total of 5 3/8 miles on Sunday and Saturday if he walked 2 and 1/4 Mile on Saturday how many did he walk on Sunday
Answers
The total miles Donovan walked were 5 3/8, we know that he walked 2 1/4 saturday then:
[tex]\begin{gathered} 5\frac{3}{8}-2\frac{1}{4}=\frac{43}{8}-\frac{9}{4} \\ =\frac{43}{8}-\frac{18}{8} \\ =\frac{25}{8} \\ =3\frac{1}{8} \end{gathered}[/tex]He walked 3 1/8 miles on sunday.
What are the solutions to the equation 3|x + 5| - 2 = 13 ?
Answers
To solve the absolute value equation;
[tex]3|x+5|-2=13[/tex]Note that the left side of the equation is an absolute value. The first step is to remove the absolute value sign, and then the next step is to solve while using the positive and negative value of the number on the right side of the equation.
This is shown below;
[tex]\begin{gathered} 3|x+5|-2=13 \\ \text{remove the absolute value sign;} \\ We\text{ now have;} \\ 3(x+5)-2=13 \\ 3(x+5)=13+2 \\ 3x+15=15 \\ \text{Subtract 15 from both sides;} \\ 3x+15-15=15-15 \\ 3x=0 \\ \text{Divide both sides by 3} \\ \frac{3x}{3}=\frac{0}{3} \\ x=0 \end{gathered}[/tex]Let us now solve for the equation when the right side is -13.
[tex]\begin{gathered} 3(x+5)-2=-13 \\ 3(x+5)=-13+2 \\ 3x+15=-11 \\ \text{Subtract 15 from both sides;} \\ 3x+15-15=-11-15 \\ 3x=-26 \\ \text{Divide both sides by 3;} \\ \frac{3x}{3}=-\frac{26}{3} \end{gathered}[/tex]ANSWER:
[tex]\begin{gathered} x=0, \\ OR \\ x=-\frac{26}{3} \end{gathered}[/tex]The last option is the correct answer
I need help writing a equation for a circleEnds of diameter: (1,-2) and (10,8)
Answers
The equation of the circle with the given end point is;
[tex](x-5.5)^2+(y-3)^2=\text{ }\frac{181}{4}[/tex]Here, we want to write the equation of a circle, given the end points of the circle
Given the end-points, we can use the formual for the distance to get the radius of the circle
We have this as follows;
[tex]\begin{gathered} D\text{ = }\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ \\ \\ (x_1,y_1)\text{ = (1,-2)} \\ (x_2,y_2)\text{ = (10,8)} \\ D\text{ = }\sqrt[]{(10-1)^2+(8-(-2))^2} \\ D\text{ = }\sqrt[]{9^2+10^2} \\ D\text{ = }\sqrt[]{181} \end{gathered}[/tex]To get the radius of the circle, we have to divide the diameter by 2
We have this as;
[tex]r\text{ = }\frac{D}{2}\text{ = }\frac{\sqrt[]{181}}{2}[/tex]Now, the other thing needed to write the equation of the circle is the center of the circle
The center of the circle can be calculated by the use of the midpoint formula
[tex]\begin{gathered} (x,y)\text{ = (}\frac{x_2+x_1}{2},\frac{y_2+y_1}{2}) \\ \\ (x,y)\text{ = (}\frac{1+10}{2},\frac{8-2}{2})\text{ = (5.5,3)} \end{gathered}[/tex]The general equation of the circle is;
[tex]\begin{gathered} (x-a)^2+(y-b)^2=r^2 \\ (a,b)\text{ = (5.5,3)} \\ r\text{ = }\frac{\sqrt[]{181}}{2} \\ \\ (x-5.5)^2+(y-3)^2=\text{ }(\frac{\sqrt[]{181}}{2})^2 \\ \\ (x-5.5)^2+(y-3)^2=\text{ }\frac{181}{4} \end{gathered}[/tex]find the slope x intercept and y intercept of the standard form equation below 7x + 3y equals 42
Answers
Given the equation of the line :
[tex]7x+3y=42[/tex]To find the slope, we will write the equation of the line in slope- intercept form
So, it will be as following :
[tex]\begin{gathered} 7x+3y=42 \\ 3y=-7x+42 \end{gathered}[/tex]Divide all terms by 3
[tex]\begin{gathered} \frac{3y}{3}=-\frac{7x}{3}+\frac{42}{3} \\ \\ y=-\frac{7}{3}x+14 \end{gathered}[/tex]Which will be similar to the general form: y = m * x + b
Where m is the slope
So, the slope of the given equation = -7/3
To find y- intercept, substitute with x = 0
[tex]\begin{gathered} 7\cdot0+3y=42 \\ 3y=42 \\ \\ y=\frac{42}{3}=14 \end{gathered}[/tex]To find x- intercept , substitute with y = 0
[tex]\begin{gathered} 7x+3\cdot0=42 \\ 7x=42 \\ \\ x=\frac{42}{7}=6 \end{gathered}[/tex]so, the answer is :
[tex]\begin{gathered} x-\text{intercept}=6 \\ y-\text{inercept}=14 \\ \text{slope}=-\frac{7}{3} \end{gathered}[/tex]
The shape below is made of two rectangles joined together.
9 cm
5 cm
8 cm
5 cm
Find the total area of the shape.
Optional working
Answers
Answer:
a=7, b=3
Step-by-step explanation:
Subtracting, we get the area of QRXY is 63 cm².
Using the area formula of a rectangle, we get 9a=63, and thus a=7.
Using the area formula again on triangle PXYS, 7b=21, and thus b=3.
The value of side 'a' is 7cm and side b is 3 cm.
What is an area?The space occupied by any two-dimensional figure in a plane is called the area. The space occupied by the rectangle in a two-dimensional plane is called the area of the rectangle.
It is given that the area of PQRS is 84 square cm and the area of PSYX is 21 square cm.
The area of QRXY is calculated as,
Area QRXY = 84 - 21
Area QRXY = 63 square cm
The side a will be calculated as,
a x 9 = 63
a = 63 / 9
a = 7 cm
The side of b will be,
b x 7 = 21
b = 21 / 7
b = 3 cm
Therefore, the value of side 'a' is 7cm and side b is 3 cm.
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Please help me dont have much time
Answers
Answer:
+, +, -4, 5, -20
Step-by-step explanation:
hope this helps
Use a suitable half-angle formula to find the exact value of cos(15°).
Answers
Remember that
[tex]\cos (\frac{x}{2})=\pm\sqrt[]{\frac{1_{}+\cos x}{2}}[/tex]For x=30 degrees
[tex]\cos (\frac{30^o}{2})=\cos (15^o)=\sqrt[]{\frac{1+\cos 30^o}{2}}[/tex]we know that
[tex]\cos 30^o=\frac{\sqrt[]{3}}{2}[/tex]substitute
[tex]\cos (15^o)=\sqrt[]{\frac{1+\frac{\sqrt[]{3}}{2}}{2}}[/tex][tex]\cos (15^o)=\sqrt[]{\frac{2+\sqrt[]{3}}{4}}[/tex][tex]\cos (15^o)=\frac{\sqrt[]{2+\sqrt[]{3}}}{2}[/tex]Function or not function (3,7), (3,6), (5,4), (4,7)
Answers
Pluto has a diameter of 1,413 miles. What does this distance equal in kilometers?
(1 mile = 1.6 kilometers)
A. 12,761 km
B. 2,275 km
C. 3,476 km
D. 143,042 km
Answers
Therefore, B (2,275 km) is correct.
Kylie has an idea for how to calculate 23 23. She says,
"Twenty times 20 is 400, and 3 times 3 is 9;, so 23 23 should be 400 plus 9, which is 409."
What is wrong with Kylie's method? Don't just start over in a different way and don't compute 23-23
directly and compare the answer with Kylie's, work with Kylie's idea. Shade the large square below,
which consists of 23 rows with 23 small squares in each row, to help you explain your answer.
Answers
Answer:
Don't just start over in a different way, work with Kylie's idea. Use the large square below, which consists of 23 rows with 23; Question: 1. Kylie has an idea for how to calculate 23-23. She says, Twenty times 20 is 400, and 3 times 3 is 9, so 23-23 should be 400 plus 9, which is 409.
Step-by-step explanation:
PUNTOS POSIBLE A parking meter that is 1.6 meters (m) tall casts a shadow 3.6 m long. At the same time, a tree casts a shadow 9 m long. 1.6 m 3.6 m 9 m What is the height of the tree?
Answers
Answer:
4 meters
Explanation:
The triangle formed by the parking meter and its shadow is similar to the triangle formed by the tree and its shadow. So, the ratio of the height of the object and its shadow is constant and we can write the following equation:
[tex]\frac{Tree}{\text{Shadow Tre}e}=\frac{\text{ Parking meter}}{Shadow\text{ Parking meter}}[/tex]So, replacing the values, we get:
[tex]\frac{Tree}{9\text{ m}}=\frac{1.6\text{ m}}{3.6\text{ m}}[/tex]Solving for the height of the tree, we get:
[tex]\begin{gathered} \frac{Tree}{9\text{ m}}\times9m=\frac{1.6\text{ m}}{3.6\text{ m}}\times9m \\ \text{Tree = 4 m} \end{gathered}[/tex]Therefore, the height of the tree is 4 meters.
Expand the sum: 3(2y-5)
Answers
The sum of the given expression is found as 6y - 15.
What is termed as the distributive property?To "distribute" something means to divide it or give a share or portion of it. The distributive property refers to the distributive law of multiplication placed above a white basic arithmetic operations such as addition and subtraction. This property states that multiplying the sum of two or even more addends by a number produces the same outcome as multiplying each addend separately by the number and afterwards adding this same products together.In other words, a combination of the type A (B+ C) can be solved using the distributive property as A (B + C) = A B +AC.For the given question,
The equation is stated as;
sum: 3(2y-5)
Using distributive property.
= 3×2y - 3×5
On simplification;
= 6y - 15
Thus the sum of the given expression is found as 6y - 15.
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The average speed, s, in miles per hour that a student walks the 3 miles from home to school varies inversely as the number of hours, h that the student walks.The formula is given by s = 3/hAs the number of hours it takes the student to walk from home to school increases, what happens to the speed?
Answers
If the number of hours it takes the student to walk home increases, then the speed decreases
Let us see an example
if h = 1
and we increase h to be = 3
[tex]\begin{gathered} \text{when h =1} \\ \text{The sp}eed,\text{ s =}\frac{3}{1}=3milesperhour \end{gathered}[/tex][tex]\begin{gathered} \text{When h increases to 3} \\ s\text{ =}\frac{3}{3}=1mileper\text{ hour} \end{gathered}[/tex]3miles/hour is more than 1 mile/hour so hence the speed will decrease when the time increases
The answer is option A, the speed decreases
4 x 107 is
?
times as much as 4 x 10³.
Answers
Step-by-step explanation:
I think thats 10 times
Answer: 4 x 10^4
Step-by-step explanation:
4 x 10^7= 40,000,000
4 x 10^3 = 4,000
Now 40,000,000 - 4,000 = 40,000
40,000 converts to 4 x 10^4
Use a graph in a [-2π, 2π, π/2] by [-3, 3, 1] viewing rectangle to complete the identity
Answers
We have the expression:
[tex]\frac{1-2\cos 2x}{2\sin x-1}[/tex]Let's work first with the numerator. We have that we can write cos2x like this:
[tex]\cos 2x=1-\sin ^2x[/tex]doing this substitution we get the following:
[tex]\begin{gathered} 1-2\cos 2x=1-2\cdot(1-\sin ^2x)=1-2+4\sin ^2x=4\sin ^2x-1 \\ =(2\sin x+1)(2\sin x-1) \end{gathered}[/tex]Now that we have these two factors, we can use them on the original identity to get:
[tex]\frac{1-2\cos2x}{2\sin x-1}=\frac{(2\sin x+1)(2\sin x-1)}{2\sin x-1}=2\sin x+1[/tex]therefore, the resulting identity is 2sinx+1
What is inverse of square root? How could the inverse be used to solve an equation that included a square root?
Answers
The power 2 (or exponent 2) is the inverse of the square root:
√x*√x = (√x)² = x
What is the inverse of the square root?
The square root of a number X, gives a number Y such that the product between Y and itself is equal to X, this means that:
if:
√x = y
Then:
y*y = x
So we could write:
√x*√x = (√x)² = x
So the exponent 2 is the inverse of the square root.
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Graph f(x)=-2x + 4. What is x when f(x)=-8? Complete the explanation on how you found x.
Answers
f(x) = - 2x + 4
To find the value of x at f(x) = -8
Substitute f(x) in the equation above by -8
-8 = -2x + 4
Subtract 4 from both sides to move 4 from the right side to the left side
-8 - 4 = -2x + 4 - 4
-12 = -2x
Divide both sides by -2 to find x
[tex]-\frac{12}{-2}=-\frac{2x}{-2}[/tex]6 = x
The value of x at f(x) = -8 is 6
The first point x = 0
f(0) = -2(0) + 4
f(0) = 0 + 4
f(0) = 4
The point is (0, 4)
The second point y = 0
f(x) = 0
0 = -2x + 4
Add 2x to both sides
0 + 2x = -2x + 2x + 4
2x = 4
Divide both sides by 2
x = 2
The point is (2, 0)
at y = -8 x = 6
Estimate the total cost of the following items for your dorm room. Round the individual cost to the nearest ten dollars. Round your answer to the nearest dollar (no cents).
Answers
First, we need to round each individual item to the nearest ten dollars.
Loft bed $149.68 = $150.00
Beanbag chair $39.73 = $40.00
Storage cubes $19.85 = $20.00
Lava lamp $19.68 = $20.00
Now, we need to add up all the items to find the total cost.
Total cost = $150.00 + $40.00 + $20.00 + $20.00
Total cost = $230
Hence, the estimated total cost for the dorm room items is $230.
Kayla's class went on a field trip to an aquarium. One tank had 30 clown fish. She miscounted the total number of clown fish in the tank and recorded it as 24 fish. What is Kayla's percent error?a 6%b 25%c 30%d 30%
Answers
Error = Value with error - Correct value
Error = 24-30 = -6
(Note tha the minus sign means the value is less than the correct value)
[tex]\text{ \%error = }\frac{\text{error}}{\text{True value}}\text{ x 100\%}[/tex][tex]=\frac{6}{30}\text{ x 100\% = 20\%}[/tex]What's the equation of the tangent to [tex]y = x^2[/tex] at [tex]x = 2[/tex]?
Answers
Answer:
y=4x-4.
Step-by-step explanation:
1. the required equation of the tangent line has the form:
y=kx+i, where k=f'(x₀), i - interception;
2. if x₀=2, then y₀=2²=4;
3. f'(x)=2x, then f'(x₀)=2*2=4;
4. the required equation with unknown 'i' is y=4x+i;
it is possible to calculate unknown 'i' using coordinates (x₀;y₀):
5. 4=4*2+i, ⇒ i=-4;
6. finally, y=4x-4.
P.S. the suggested solution is not the shortest one.
Express y in terms of x. Then find the value of y when x= -15 - y = 3xY in terms of x:Y=
Answers
first solve y
[tex]\begin{gathered} 5-y=3x \\ 5-y-5=3x-5 \\ -y=3x-5 \\ (-1)\times-y=(-1)\times(3x-5) \\ y=-3x+5 \end{gathered}[/tex]now replace x=-1
[tex]\begin{gathered} y=-3(-1)+5 \\ y=3+5 \\ y=8 \end{gathered}[/tex]Use a net to find the surface area of the rectangular prism the height of the rectangular prism meets the base at a 90° angle
Answers
Answer:
406 square inches
Explanation:
The net of a rectangular prism has 6 rectangular faces.
In this prism, the dimensions of the faces are:
• 2 rectangles with length 7 in. and width 5 in.
,• 2 rectangles with length 7 in. and width 14 in.
,• 2 rectangles with length 14 in. and width 5 in.
Next, we find the surface area of the prism:
[tex]\begin{gathered} \text{Surface Area}=2(7\times5)+2(7\times14)+2(14\times5) \\ =2(35)+2(98)+2(70) \\ =70+196+140 \\ =406in^2 \end{gathered}[/tex]The surface area of the rectangular prism is 406 in².
Solve for u.
(u-1)²=2u²+8u+25
Answers
Answer:
u = -4 or -6
Step-by-step explanation:
We know that,
( a + b ) = a² + 2ab + b²
Accordingly, let us solve the given equation.
( u - 1 )² = 2u² + 8u + 25
First, solve the brackets.
u² - 2u + 1 = 2u² + 8u + 25
Subtract u² by both sides.
-2u + 1 = 2u² - u² + 8u + 25
-2u + 1 = u² + 8u + 25
Now, add 2u by both sides.
1 = u² + 8u + 2u + 25
Subtract 1 by both sides.
0 = u² + 10u + 25 -1
0 = u² + 10u + 24
And solve the quadratic equation and solve for u.
0 = u² + 6u + 4u + 24
0 = u ( u + 6 ) + 4 ( u + 6 )
0 = ( u + 4 ) ( u + 6 )
Therefore,
u + 4 = 0
u = -4
or
u + 6 = 0
u = -6
Andre used 1/2 of a stick of butter to make multiple batches of brownies. The recipe calls 1/8 for of a stick of butter for each batch. How many batches did he make?
Please help!! Will give brainlest
Answers
1/8 + 1/8 + 1/8 + 1/8 = 4/8
4/8 = 1/2
A right triangle is formed by the x-axis, the y-axis, and the line y = -2x + 7.A. Sketch and label a graph of the triangle in the coordinate plane.B. Find the length of the hypotenuse.C. Find the area of the triangle.
Answers
The given equation is
[tex]y=-2x+7[/tex]We have to find the axis interceptions to graph this function.
For x = 0, we have
[tex]y=-2(0)+7=0+7=7[/tex]The y-interception is (0,7).
For y = 0, we have
[tex]\begin{gathered} 0=-2x+7 \\ 2x=7 \\ x=\frac{7}{2} \end{gathered}[/tex]The x-interception is (7/2, 0).
Now, we graph.
Where h is the hypothenuse.
To find the hypothenuse of the right triangle formed, we use the distance formula and both points.
[tex]d=\sqrt[]{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]Replacing the coordinates, we have
[tex]\begin{gathered} d=\sqrt[]{(0-7)^2+(3.5-0)^2}=\sqrt[]{(-7)^2+(3.5)^2} \\ d=\sqrt[]{49+12.25}=\sqrt[]{61.25}\approx7.83 \end{gathered}[/tex]Therefore, the hypotenuse is 7.83 units long, approximately.
At last, the area of the triangle is found with the formula
[tex]A=\frac{1}{2}b\cdot h[/tex]Where b is the base and h is the height of the triangle. b = 3.5, h = 7.
Replacing these values, we have
[tex]A=\frac{1}{2}(3.5)\cdot(7)=\frac{24.5}{2}=12.25u^2[/tex]Therefore, the area of the triangle is 12.25 square units.
I tried to solve this but I immediately got confused.
Answers
Firstly, let us find the perimeter.
Perimeter P is;
[tex]P=AB+BC+AC[/tex]The length of AB and AC can be calculated using the formula for distance between two points.
[tex]\begin{gathered} AB=\sqrt[]{4^2+2^2} \\ AB=\sqrt[]{16+4} \\ AB=\sqrt[]{20} \\ AB=4.47\operatorname{cm} \end{gathered}[/tex][tex]\begin{gathered} AC=\sqrt[]{4^2+4^2} \\ AC=\sqrt[]{16+16} \\ AC=\sqrt[]{32} \\ AC=5.66\operatorname{cm} \end{gathered}[/tex][tex]BC=6\operatorname{cm}[/tex]substituting we have;
[tex]\begin{gathered} P=AB+BC+AC \\ P=4.47\operatorname{cm}+6\operatorname{cm}+5.66\operatorname{cm} \\ P=16.13\operatorname{cm} \end{gathered}[/tex]The perimeter is 16.13 cm.
Secondly, The Area A.
[tex]A=\frac{1}{2}bh[/tex][tex]undefined[/tex]What number is best to use to simplify a fraction Least common multiple be least common denominator see greatest common factor de greatest
Answers
The greatest common factor and least common multiple are the best used to simplify a fraction.
Suppose we wanted to add the fractions:
10/12 and 20/15.
Now, consider the fraction:
10/12
We can eliminate that component from both the numerator and the denominator and simplify the fraction if we can identify the greatest common factor between 10 and 12.
So,
The factors of 10 are 1, 2, 5, 10.
The factors of 12 are 1, 2, 3, 4, 6, 12.
Therefore, GCF of 10 and 12 is 2.
Hence,
10/12 = ( 10 ÷ 2 ) / ( 12 ÷ 2 ) = 5/6
Similarly,
For 20/15,
The factors of 20 are 1, 2, 4, 5, 10, 20.
The factors of 15 are 1, 3, 5, 15.
The greatest common factor is 5.
So,
20/15 = ( 20 ÷ 5 ) / ( 15 ÷ 5 ) = 4/3
Now, when we add the fractions:
10/12 + 20/15 = 5/6 + 4/3
Now, as the fractions are already simplified.
By using the least common multiple.
10/12 + 20/15 = 5/6 + ( 4/3 ) × ( 2/2 )
10/12 + 20/15 = 5/6 + 8/6
10/12 + 20/15 = 13/6
Hence, using the greatest common factor we can simplify the fractions, and the least common multiple we can perform operations on the fractions.
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The maximum grade allowed between two stations in a rapid transit rail system is 3.6% between station A and station , which are 300 feet apart, the tracks rise 7ft. What is the grade of the tracks between these stations? round the answer to the nearest tenth of a percent. Does this grade meet the rapid-transit rail standards
Answers
To determine the grade we first need to find the slope of the track, the slope is given by:
[tex]slope=\frac{rise}{run}[/tex]In this case, we know that the track rise 7 ft in a 300 ft run, then we have:
[tex]slope=\frac{7}{300}=0.023[/tex]Now that we know the slope we just multiply it by 100 to find the grade:
[tex]\begin{gathered} grade=100*(0.023) \\ grade=2.3 \end{gathered}[/tex]Hence the grade of the track is 2.3% and since this is less than 3.6% we conclude that this track meets the rapid-transit rail standards
Anton counted 36 dogs and cats at the animal shelter. He noticed that there are 18 fewer cats than dogs. If c= the number of cats and d= the number of dogs, then which system of equations represents the situation?
Answers
Answer: 2d + 18 = 36
Step-by-step explanation:
c = number of cats, d = number of dogs
1. d + c = 36 because there are 36 cats and dogs combined
2. c = d - 18 because there are 18 fewer cats than dogs
3. d + d - 18 = 36 because we inserted the equation of step 2 in the equation of step 1
4. 2d + 18 = 36 is the simplified equation of step 3
a rectangle has a lenght of 33 yard less than 6 times it's width. if the area of the rectangle is 6195 square yards, find the length of the rectangle
Answers
Answer
The length of the Rectangle is 177 yards
SOLUTION
Problem Statement
The question tells us that the rectangle has a length of 33 yards less than 6 times its width. We are asked to find the length of the rectangle given that the area of the rectangle is 6195 square yards.
Solution
To solve the question, we simply need to interpret each sentence of the question
Let us go through each portion and come up with equations.
[tex]\begin{gathered} Let\text{ the length of the rectangle be }l \\ \text{Let the width of the rectangle be }w \\ \\ \text{ We are told the length is 33 yards times less than 6 times its width: } \\ l=6w-33\text{ (Equation 1)} \\ \\ \text{ We are told that the Area of the rectangle is 6196} \\ \therefore l\times w=6195\text{ (Equation 2)} \end{gathered}[/tex]Now that we have the equations, we can solve them simultaneously. We shall use the substitution method.
[tex]\begin{gathered} l=6w-33\text{ (Equation 1)} \\ lw=6195\text{ (Equation 2)} \\ \text{From Equation 2, we have that:} \\ w=\frac{6195}{l} \\ \text{ Substituting the expression for w into Equation 1.} \\ \\ l=6w-33\text{ becomes,} \\ l=6(\frac{6195}{l})-33 \\ \text{ Multiply both sides by l} \\ l\times l=l(\frac{6\mleft(6195\mright)}{l}-33) \\ \\ l^2=37,170-33l \\ \text{ rewrite the equation, we have:} \\ l^2+33l-37170=0 \end{gathered}[/tex]
We have obtained a quadratic equation in terms of the length of the rectangle. After solving the equation, we can find the length of the rectangle.
To solve, we shall apply the Quadratic Formula. The Quadratic Formula is given by:
[tex]\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \text{Given the Quadratic equation,} \\ ax^2+bx+c=0 \\ (In\text{ our case, x is }l) \end{gathered}[/tex]Let us apply the formula to our equation as follows:
[tex]\begin{gathered} Given\text{ the equation: } \\ l^2+33l-37170=0 \\ a=1,b=33,c=-37170 \\ \\ \therefore l=\frac{-33\pm\sqrt[]{33^2-4(-37170)(1)}}{2(1)} \\ \\ l=\frac{-33\pm\sqrt[]{1089+148,680}}{2} \\ \\ l=\frac{-33\pm\sqrt[]{149,769}}{2} \\ \\ l=\frac{-33\pm387}{2} \\ \\ l=177\text{ or -210} \\ \\ \text{ Since we are dealing with lengths and lengths cannot be negative, } \\ \text{Length of the Rectangle is 177 yards} \end{gathered}[/tex]Final Answer
The length of the Rectangle is 177 yards
