GEOMETRY 100 POINTS CHALLENGE ​​

well, if we take a peek at the number line above, from one integer to the next there are 5 slots or divisions, meaning each division is 1/5.

what's the distance of DE?

well, D is at -1 plus 4 slots, that means -1 and we move it further by 4/5, that makes it -1⅘.

now E is at 2 plus 2 slots more, so that means 2 and we move it further by 2/5 or 2⅖.

to get the distance we simply get their difference, not much to it.

[tex]\stackrel{mixed}{1\frac{4}{5}}\implies \cfrac{1\cdot 5+4}{5}\implies \stackrel{improper}{\cfrac{9}{5}}~\hfill \stackrel{mixed}{2\frac{2}{5}} \implies \cfrac{2\cdot 5+2}{5} \implies \stackrel{improper}{\cfrac{12}{5}} \\\\[-0.35em] ~\dotfill\\\\ |DE|\implies \left| -\cfrac{9}{5}-\cfrac{12}{5} \right|\implies \left| -\cfrac{21}{5} \right|\implies \cfrac{21}{5}\implies \text{\LARGE 4.2}[/tex]

how about the midpoint of DE?

well, we can move from the left over to the midpoint or we can move from  the right, let's move from the  right, so hmmm let's take half of 21/5 and subtract it from E, that's where the midpoint is.

[tex]\stackrel{ E }{\cfrac{12}{5}}~~ - ~~\stackrel{ \textit{half of DE} }{\left( \cfrac{21}{5}\cdot \cfrac{1}{2} \right)}\implies \cfrac{12}{5}-\cfrac{21}{10}\implies \cfrac{(2)12~~ - ~~(1)21}{\underset{\textit{using this LCD}}{10}} \\\\\\ \cfrac{24-21}{10}\implies \cfrac{3}{10}\implies \text{\LARGE 0.3}[/tex]

Sophia will drive a distance of 300 miles.

To find out how far Sophia will drive, we need to calculate 3/5 of the total trip distance of 500 miles.

1: Calculate the distance driven by Sophia

To find the distance driven by Sophia, we need to multiply the total trip distance by the fraction representing the portion she will drive. Sophia will drive the first 3/5 of the trip.

Distance driven by Sophia = (3/5) * 500 miles

2: Simplify the fraction

To calculate the distance, we simplify the fraction 3/5.

Distance driven by Sophia = (3/5) * 500 miles

                       = (3 * 500) / 5 miles

                       = 1500 / 5 miles

                       = 300 miles

Therefore, Sophia will drive a distance of 300 miles.

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Step-by-step explanation:

Since it's five (5) sided it is a pentagon.

First find the sum of the interior angle of a Pentagon .

S = ( n -2 ) × 180°

n represent the number of sides

S = ( 5 -2 ) × 180°

S = 3×180°

S = 540°

5x+2+ 10x-3 +7x-11 +8x-19 + 13x -31 = 540°

43x - 62 = 540°

43x = 540° + 62

43x = 602°

x = 14°

Answer : 14°

Answer:

1) x = 43.33

2) x =  tan⁻¹(4/3)  = 53.13

Step-by-step explanation:

1) Formula: cos(A) = sin(90 - A)

sin(2x - 30) = cos(x - 10)

⇒ sin(2x - 30) = sin(90 - (x - 10))

⇒ sin(2x - 30) = sin(90 - x + 10)

⇒ sin(2x - 30) = sin(100 - x)

⇒ 2x - 30 = 100 - x

⇒ 3x = 130

⇒ x = 130/3

x = 43.33

2) 4sinx . cosx - 3sin²x = 0

⇒ 4sinx . cosx = 3sin²x

⇒ 4cosx = 3sinx

⇒ [tex]\frac{4}{3} =\frac{sinx}{cosx}[/tex]

⇒ [tex]tan x = \frac{4}{3}[/tex]

⇒ x =  tan⁻¹(4/3)

⇒ x = 53.13

Answer:

[tex]\textsf{1)} \quad x \approx 43.33^{\circ}[/tex]

[tex]\textsf{2)} \quad \boxed{\begin{aligned}x &= \pi n \;\text{radians}\\x&=0.93+\pi n\; \text{radians}\end{aligned}}\quad \boxed{\begin{aligned}x &= 180^{\circ}n\\x&=53.13^{\circ}+180^{\circ} n\; \end{aligned}}[/tex]

Step-by-step explanation:

Given trigonometric equation:

[tex]\sin(2x - 30^{\circ}) = \cos(x - 10^{\circ})[/tex]

To solve the given trigonometric equation, we can use the following trigonometric identity:

[tex]\boxed{\begin{minipage}{4 cm}\underline{Trigonometric identity} \\\\$\cos (\theta)=\sin(90^{\circ}-\theta)$\\\end{minipage}}[/tex]

Apply the trigonometric identity to the right side of the equation:

[tex]\begin{aligned}\sin(2x - 30^{\circ}) &= \cos(x - 10^{\circ})\\\\&= \sin(90^{\circ}-(x - 10^{\circ}))\\\\&= \sin(90^{\circ}-x +10^{\circ})\\\\&= \sin(100^{\circ}-x)\end{aligned}[/tex]

Since the sine function is equal, we can equate the angles:

[tex]2x - 30^{\circ}=100^{\circ}-x[/tex]

Now simplify and solve for x:

[tex]\begin{aligned}2x - 30^{\circ}&=100^{\circ}-x\\\\2x - 30^{\circ}+x&=100^{\circ}-x+x\\\\3x - 30^{\circ}&=100^{\circ}\\\\3x - 30^{\circ}+30^{\circ}&=100^{\circ}+30^{\circ}\\\\3x&=130^{\circ}\\\\\dfrac{3x}{3}&=\dfrac{130^{\circ}}{3}\\\\x&=\left(\dfrac{130}{3}\right)^{\circ}\\\\x&\approx 43.3^{\circ}\; \sf (nearest\;tenth)\end{aligned}[/tex]

Therefore, the solution to the equation sin(2x - 30°) = cos(x - 10°) is approximately x = 43.33°.

[tex]\hrulefill[/tex]

Given trigonometric equation:

[tex]4 \sin x \cos x-3\sin^2x=0[/tex]

Factor out the common term sin(x):

[tex]\sin x(4 \cos x-3\sin x)=0[/tex]

According to the zero product property, one of the factors must be equal to zero for the equation to hold.

Set each factor equal to zero and solve for x.

Factor 1

[tex]\sin x=0[/tex]

According to the unit circle, sin(x) = 0 when x = 0 and x = π.

As the sine function is periodic with a period of 2π, the solutions to sin(x) = 0 are:

[tex]x=0+2\pi n, \;\;x=\pi + 2\pi n[/tex]

Therefore, x is any multiple of π, where n is an integer:

[tex]\boxed{x = \pi n}[/tex]

Factor 2

[tex]\begin{aligned}4\cos x - 3 \sin x & = 0\\\\4 \cos x & = 3 \sin x\\\\\dfrac{4}{3}&=\dfrac{\sin x}{\cos x}\\\\\dfrac{4}{3}&=\tan x\\\\\implies x&=\arctan\left(\dfrac{4}{3}\right)\\\\x&=0.92729...\end{aligned}[/tex]

As the tangent function is periodic with a period of π, the solutions are:

[tex]\boxed{x=0.92729...+\pi n}[/tex]

where n is an integer.

Therefore, the solutions to the equation 4sin(x)cos(x) - 3sin²(x) = 0 are:

[tex]\boxed{\begin{aligned}x &= \pi n \;\text{radians}\\x&=0.93+\pi n\; \text{radians}\end{aligned}}[/tex]    [tex]\boxed{\begin{aligned}x &= 180^{\circ}n\\x&=53.13^{\circ}+180^{\circ} n\; \end{aligned}}[/tex]

(where n is an integer)

i) 4 more gallons of petrol will be needed to complete the journey.

ii) The cost of the petrol for the 420 km journey is GH¢55.00.

i) To determine the number of gallons of petrol needed to complete the journey, we can calculate the total distance that can be covered with the available petrol and then subtract it from the total distance of the journey.

Given that the car consumes 1 gallon of petrol for every 30 km, we can calculate the distance that can be covered with 10 gallons of petrol by multiplying 10 (gallons) by 30 (km/gallon):

Distance covered with 10 gallons = 10 * 30 = 300 km

To find the remaining distance that needs to be covered, we subtract the distance covered with the available petrol from the total distance of the journey:

Remaining distance = Total distance - Distance covered with available petrol

Remaining distance = 420 km - 300 km = 120 km

Since the car consumes 1 gallon of petrol for every 30 km, we can determine the additional gallons of petrol needed by dividing the remaining distance by 30:

Additional gallons needed = Remaining distance / 30 = 120 km / 30 km/gallon = 4 gallons

Therefore, the driver will need 4 more gallons of petrol to complete the journey.

ii) To calculate the cost of the petrol for the journey of 420 km, we need to multiply the total number of gallons used for the journey by the cost per gallon.

Given that a gallon of petrol costs GH¢5.50, and the total number of gallons used for the journey is 10 (given in the problem), we can calculate the cost using the formula:

Cost of petrol = Total gallons used * Cost per gallon

Cost of petrol = 10 gallons * GH¢5.50/gallon = GH¢55.00

Therefore, the cost of the petrol for the journey of 420 km is GH¢55.00.

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Part (a)

Answer: Constant of proportionality = 5/8

Reason:

The general template equation is y = kx where k is the constant of proportionality. It is the slope of the line.

The direct proportion line must pass through the origin. In other words, the y intercept must be zero.

=====================================

Part (b)

Answer: Not Proportional

Reason:

The y intercept isn't zero.

Plug x = 0 into the equation to find y = 1 is the y intercept. This graph does not pass through the origin.

A shirt is priced at $16, and a sweater carries a price tag of $33.

To solve this problem, we can set up a system of equations based on the given information. Let's represent the cost of a shirt as 's' and the cost of a sweater as 'w'.

From the first statement, Andrea is able to buy 3 shirts and 2 sweaters for $114. We can express this as the equation:

3s + 2w = 114.

From the second statement, Andrea is able to buy 2 shirts and 4 sweaters for $164. We can express this as the equation:

2s + 4w = 164.

To solve this system of equations, we can use the method of substitution or elimination. Here, let's use the substitution method.

First, let's solve the first equation for 's':

3s + 2w = 114

3s = 114 - 2w

s = (114 - 2w)/3.

Now, substitute the expression for 's' in the second equation:

2((114 - 2w)/3) + 4w = 164

(228 - 4w)/3 + 4w = 164

228 - 4w + 12w = 492

8w = 492 - 228

8w = 264

w = 264/8

w = 33.

Now, substitute the value of 'w' into the first equation to find the value of 's':

3s + 2(33) = 114

3s + 66 = 114

3s = 114 - 66

3s = 48

s = 48/3

s = 16.

Therefore, a shirt costs $16, and a sweater costs $33.

In summary, based on the given information and the solution to the system of equations, the cost of a shirt is $16, and the cost of a sweater is $33.

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Answer:

The selling price of the diamond necklace at Shamin Jewelers after 10% discount is:

$442 * 0.9 = $397.80

The selling price of the same necklace at the competitor's store after 34% and 14% discount is:

$527 * 0.66 * 0.86 = $247.08

So, Shamin Jewelers needs to offer an additional discount to meet the competitor's price:

$397.80 - $247.08 = $150.72

To calculate the additional rate of discount, we divide the difference by the original selling price at Shamin Jewelers and multiply by 100:

($150.72 / $442) * 100 = 34.11%

Therefore, Shamin Jewelers must offer an additional 34.11% discount to meet the competitor's price.

Step-by-step explanation:

Answer:

y=10

Step-by-step explanation:

y=3x+4

substitute for x

y=3(2)+4

y=6+4

y=10

Answer:

x- intercept = - 8 , y- intercept = 4

Step-by-step explanation:

to find the x- intercept let y = 0 in the equation and solve for x

x - 2(0) = - 8

x - 0 = - 8

x = - 8 ← y- intercept

to find the y- intercept let x = 0 in the equation and solve for y

0 - 2y = - 8

- 2y = - 8 ( divide both sides by - 2 )

y = 4 ← y- intercept

The complete frequency table is

Interval          Tally     Frequency

0 - 1                 |||||           5

2 - 3                               0

4 - 5                 ||||           4

6 - 7                 ||||||||         8

8 - 9                 ||||||          6

From the question, we have the following parameters that can be used in our computation:

The set of data

The tally is used to calculate the frequencies of the readings using marks

So, we have

Interval          Tally     Frequency

0 - 1                 |||||           5

2 - 3                               0

4 - 5                 ||||           4

6 - 7                 ||||||||         8

8 - 9                 ||||||          6

The above represents the complete frequency table

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Answer:

x = 16

Step-by-step explanation:

Opposite sides are equal in a parallelogram

AD = BC

5x - 12 = 3x + 20

5x - 3x = 20 + 12

2x = 32

x = 32/2

x = 16

Answer:

arc PN = 136°

Step-by-step explanation:

the inscribed angle PLN is half the measure of its intercepted arc PN

then PN is twice the inscribed angle PLN , that is

PN = 2 × 68° = 136°

The measure of the angle m∠DAX on the straight line CX is equal to 140°

Angles that are on a straight line involves the sum of angles that can be arranged together so that they form a straight line. Angles on a straight line when added together sum up to 180°.

Given that AW bisects the angle m∠CAD and angle m∠CAW is equal to 20°, then;

m∠CAD = 2 × 20°

m∠CAD = 40°

m∠DAX + m∠CAD = 180° {sum of angles on a straight line}

m∠DAX + 40° = 180°

m∠DAX = 180° - 40° {subtract 40° from both sides}

m∠DAX = 140°

Therefore, the measure of the angle m∠DAX on the straight line CX is equal to 140°

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Answer: The measure of the angle m∠DAX on the straight line CX is equal to 140°

Step-by-step explanation: the gut below beat me to it

Answer:

x = 10

Step-by-step explanation:

The diagonals of a rectangle are equal in length and bisect each other

⇒ JL = KM and

JN = LN = KN =MN

ΔJMN is an isosceles triangle since JN = MN

This means that the angles opposite to equal sides are equal

⇒ ∠JMN = ∠MJN

⇒ 7x - 2 = 3x + 38

⇒ 7x - 3x = 38 + 2

⇒ 4x = 40

⇒ x = 10

Answer:

Step-by-step explanation:

Answer:

---------------

Let the numbers be s and l.

We are given that:

Set up equations:

Eliminate l:

Solve for s:

Find l:

Step-by-step explanation:

A) The slope between the points (0, -4) and (2, -1) is 1.5.

b)  the slope between the points (0, -4) and (4, -1) is 0.75.

c) The slope of 0.75 indicates that for every 1 unit increase in x, there is a 0.75 unit increase in y.

Part A: To calculate the slope between two points, we use the formula:

slope = (change in y) / (change in x)

Let's choose the points (0, -4) and (2, -1).

Change in y = -1 - (-4) = 3

Change in x = 2 - 0 = 2

slope = 3 / 2 = 1.5

Therefore, the slope between the points (0, -4) and (2, -1) is 1.5.

Part B: Let's choose different points from the table, such as (0, -4) and (4, -1).

Change in y = -1 - (-4) = 3

Change in x = 4 - 0 = 4

slope = 3 / 4 = 0.75

Thus, the slope between the points (0, -4) and (4, -1) is 0.75.

Part C: The slopes from parts A and B provide information about the relationship between the points.

In part A, where the slope was calculated as 1.5, we can see that as the x-values increase by 2 units, the y-values increase by 3 units. This suggests a positive relationship between the points, meaning that as x increases, y also increases. The slope of 1.5 indicates that for every 1 unit increase in x, there is a 1.5 unit increase in y.

In part B, where the slope was calculated as 0.75, we observe a similar positive relationship between the points. As the x-values increase by 4 units, the y-values increase by 3 units. The slope of 0.75 indicates that for every 1 unit increase in x, there is a 0.75 unit increase in y.

Overall, the positive slopes in both parts A and B suggest that the points on the table exhibit a positive linear relationship. This means that as x increases, y also increases, and the rate of change is consistent based on the slope.

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The sets of side measurements that can form a triangle are: c. 8, 3, 11

d. 5, 11, 8

To determine which set of side measurements could be used to form a triangle, we need to apply the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let's evaluate each set of side measurements:

a. 12, 23, 7

In this case, the sum of the lengths of the two shorter sides (12 and 7) is 19, which is less than the length of the longest side (23). Therefore, this set of side measurements cannot form a triangle.

b. 10, 7, 2

Here, the sum of the two shorter sides (10 and 2) is 12, which is greater than the length of the longest side (7). This set of side measurements also fails to satisfy the triangle inequality theorem and cannot form a triangle.

c. 8, 3, 11

The sum of the two shorter sides (8 and 3) is 11, which is greater than the length of the longest side (11). Therefore, this set of side measurements can form a triangle.

d. 5, 11, 8

In this case, the sum of the lengths of the two shorter sides (5 and 8) is 13, which is greater than the length of the longest side (11). Hence, this set of side measurements can also form a triangle.

In summary, the sets of side measurements that can form a triangle are:

c. 8, 3, 11

d. 5, 11, 8

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Answer:

[tex]\displaystyle \frac{75}{2}[/tex] or [tex]37.5[/tex]

Step-by-step explanation:

We can answer this problem geometrically:

[tex]\displaystyle \int^6_{-4}f(x)\,dx=\int^1_{-4}f(x)\,dx+\int^3_1f(x)\,dx+\int^6_3f(x)\,dx\\\\\int^6_{-4}f(x)\,dx=(5*5)+\frac{1}{2}(2*5)+\frac{1}{2}(3*5)\\\\\int^6_{-4}f(x)\,dx=25+5+7.5\\\\\int^6_{-4}f(x)\,dx=37.5=\frac{75}{2}[/tex]

Notice that we found the area of the rectangular region between -4 and 1, and then the two triangular areas from 1 to 3 and 3 to 6. We then found the sum of these areas to get the total area under the curve of f(x) from -4 to 6.

Answer:

[tex]\dfrac{75}{2}[/tex]

Step-by-step explanation:

The value of a definite integral represents the area between the x-axis and the graph of the function you’re integrating between two limits.

[tex]\boxed{\begin{minipage}{8.5 cm}\underline{De\:\!finite integration}\\\\$\displaystyle \int^b_a f(x)\:\:\text{d}x$\\\\\\where $a$ is the lower limit and $b$ is the upper limit.\\\end{minipage}}[/tex]

The given definite integral is:

[tex]\displaystyle \int^6_{-4} f(x)\; \;\text{d}x[/tex]

This means we need to find the area between the x-axis and the function between the limits x = -4 and x = 6.

Notice that the function touches the x-axis at x = 3.

Therefore, we can separate the integral into two areas and add them together:

[tex]\displaystyle \int^6_{-4} f(x)\; \;\text{d}x=\int^3_{-4} f(x)\; \;\text{d}x+\int^6_{3} f(x)\; \;\text{d}x[/tex]

The area between the x-axis and the function between the limits x = -4 and x = 3 is a trapezoid with bases of 5 and 7 units, and a height of 5 units.

The area between the x-axis and the function between the limits x = 3 and x = 6 is a triangle with base of 3 units and height of 5 units.

Using the formulas for the area of a trapezoid and the area of a triangle, the definite integral can be calculated as follows:

[tex]\begin{aligned}\displaystyle \int^6_{-4} f(x)\; \;\text{d}x & =\int^3_{-4} f(x)\; \;\text{d}x+\int^6_{3} f(x)\; \;\text{d}x\\\\& =\dfrac{1}{2}(5+7)(5)+\dfrac{1}{2}(3)(5)\\\\& =30+\dfrac{15}{2}\\\\& =\dfrac{75}{2}\end{aligned}[/tex]

Answer:

x = 6

∠D = 127

Step-by-step explanation:

In a parallelogram, opposite angles are equal and adjacent angles add to 180

⇒ ∠A = ∠C and ∠C + ∠D = 180

∠A = ∠C

⇒ 13x - 25 = 9x - 1

⇒ 13x - 9x = 25 - 1

⇒ 4x = 24

⇒ x = 24/4

⇒ x = 6

∠C =  9x - 1

= 9(6) - 1

= 54 - 1

= 53

∠C + ∠D = 180

⇒ ∠D = 180 - ∠C

= 180 - 53

= 127

Answer:

p which is the y axis is -1

Hello!

area

= 2*25 + (20 - 2)*(25-8)

= 50cm² + 306cm²

= 356cm²

The expansion of the function [tex](3x - 4)^4[/tex] simplifies to [tex]81x^4 - 432x^3 + 864x^2 - 768x + 256.[/tex]

To expand the function [tex]f(x) = (3x - 4)^4[/tex], we can use the binomial theorem. According to the binomial theorem, for any real numbers a and b and a positive integer n, the expansion of [tex](a + b)^n[/tex] can be written as:

[tex](a + b)^n = C(n, 0)a^n b^0 + C(n, 1)a^{(n-1)} b^1 + C(n, 2)a^{(n-2)} b^2 + ... + C(n, n-1)a^1 b^{(n-1)} + C(n, n)a^0 b^n[/tex]

where C(n, k) represents the binomial coefficient, which is given by C(n, k) = n! / (k!(n-k)!).

Applying this formula to our function [tex]f(x) = (3x - 4)^4[/tex], we have:

[tex]f(x) = C(4, 0)(3x)^4 (-4)^0 + C(4, 1)(3x)^3 (-4)^1 + C(4, 2)(3x)^2 (-4)^2 + C(4, 3)(3x)^1 (-4)^3 + C(4, 4)(3x)^0 (-4)^4[/tex]

Simplifying each term, we get:

[tex]f(x) = 81x^4 + (-432x^3) + 864x^2 + (-768x) + 256[/tex]

Therefore, the expanded form of the function [tex]f(x) = (3x - 4)^4[/tex] is [tex]81x^4 - 432x^3 + 864x^2 - 768x + 256[/tex].

Note that the coefficient of [tex]x^3[/tex] is -432, the coefficient of [tex]x^2[/tex] is 864, the coefficient of x is -768, and the constant term is 256.

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Note the complete question is

The calculated ratio of the flowers are

From the question, we have the following parameters that can be used in our computation:

Roses = 5

Flowers = 12

This means that

Daises = 12 - 5

Evaluate

Daises = 7

Next, we have the ratio to be

Daises : Roses = 7 : 5

Also, we have

Flower : Daises = 12 : 5

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The park's attendance on the 30th day of June will be significantly higher than on the 31st day of August.

The amusement park in question has a starting attendance of 2000 guests on the first day of both June and August.

The park's attendance in June increases by 10% each day, while its attendance in August decreases by 10% each day.

The rate of change of the attendance in June and August can be calculated as follows:

Let's assume the attendance on the first day in June is equal to A.

Therefore, the attendance in June can be calculated as follows: On the 2nd day, the attendance would be 1.1A.

On the 3rd day, the attendance would be 1.1 × 1.1A = 1.21A

On the 4th day, the attendance would be 1.1 × 1.1 × 1.1A = 1.331A

And so on, until the 30th day of June.A = 2000

Therefore, on the 30th day of June, the attendance will be equal to:

Attendance on the 30th day = 2000 × 1.1^29 = 20,062.46 guests (rounded to the nearest whole number)

The attendance in August can be calculated in the same way, except that the rate of change will be negative (i.e., a decrease of 10% per day):

Attendance on the 2nd day = 0.9A

Attendance on the 3rd day = 0.9 × 0.9A = 0.81A

Attendance on the 4th day = 0.9 × 0.9 × 0.9A = 0.729A

And so on, until the 31st day of August.A = 2000

Therefore, on the 31st day of August, the attendance will be equal to: Attendance on the 31st day = 2000 × 0.9^30 = 49.28 guests (rounded to the nearest whole number)

Therefore, the park's attendance on the 30th day of June will be significantly higher than on the 31st day of August.

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Answer:

x = 3/2 or x = -1

Step-by-step explanation:

2x² - x - 3 = 0

2*(-3) = -6

Factors of -6:

(-1, 6), (1, -6), (-2, 3), (2, -3)

We need to find a pair that adds up to the co-eff of x which is (-1)

Factors :(2,-3)

2 - 3 = -1

so, 2x² - x - 3 = 0 can be written as:

2x² + 2x - 3x - 3 = 0

⇒ 2x(x + 1) -3(x + 1) = 0

⇒ (2x - 3)(x + 1) = 0

⇒ 2x - 3 = 0 or

x + 1 = 0

⇒ 2x = 3 or x = -1

⇒ x = 3/2 or x = -1