Solved Examples and Worksheet for Solving Quadratic Equations Graphically

Q1Estimate the solution of 2x2 - x = 3 using the graph.


A. 1 and -2
B. 2 and 8
C. -1 and 3
D. -1 and 32

Solutions
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Step: 1
From the figure, the graph intersects the x - axis at two points (-1, 0) and ( 32, 0).
Step: 2
The x-intercepts of the curve are -1 and 32.
Step: 3
2x2 - x = 3
  [Original equation]
Step: 4
2(-1)2 - (-1) = 3
  [Substitute x-intercept as -1.]
Step: 5
3 = 3
  [Simplify.]
Step: 6
2( 32)2 - (32) = 3
  [Substitute x-intercept as 32.]
Step: 7
3 = 3
  [Simplify.]
Step: 8
Both the values satisfy the equation. So, -1 and 32 are the solutions of the equation.
Correct Answer is :   -1 and 32
Q2Find the solution of - 3x2 = - 27 using a graph.
A. - 3 and 6
B. - 3 and 7
C. - 4 and 4
D. - 3 and 3

Solutions
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Step: 1
- 3x2 = - 27
  [Original equation]
Step: 2
x2 = 9
  [Divide - 3 on each side]
Step: 3
x2 - 9 = 9 - 9
  [Subtract 9 from each side]
Step: 4
x2 - 9 = 0
  [Simplify.]
Step: 5
Sketch the graph of the related quadratic function y = x2 - 9 as shown below.
Step: 6
From the graph, the x-intercepts appear to be - 3 and 3.
  [Estimate the values of the x-intercepts.]
Step: 7
So, - 3 and 3 are the solutions of the equation.
Correct Answer is :   - 3 and 3
Q3A rectangular fountain in a park has dimensions of a and a + 16. If the area of the fountain is 192 square meters, find the dimensions of the fountain in meters.
A. 9, 25
B. 11, 27
C. 8, 24
D. 10, 26

Solutions
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Step: 1
The area of a rectangle = length × width.
Step: 2
The area of the rectangular fountain = (a) × (a + 16) square meters.
Step: 3
192 = (a) × (a + 16)
  [Original equation.]
Step: 4
192 = a2 + 16a
  [Use distributive property.]
Step: 5
192 + 82 = a2 + 16a + 82
  [Add (162)2 = 82 = 64 to each side.]
Step: 6
256 = (a + 8)2
  [Write the right hand side as a perfect square and simplify.]
Step: 7
± 16 = a + 8
  [Evaluate square roots on both sides.]
Step: 8
± 16 - 8 = a + 8 - 8
  [Subtract 8 from each side.]
Step: 9
a = + 8 or - 24
  [Simplify.]
Step: 10
Width = a = 8 meters
  [The dimensions cannot be negative.]
Step: 11
Length = (a + 16) = (16 + 8) = 24 meters.
  [Substitute 8 for a and add.]
Step: 12
The dimensions of the fountain are 8 meters wide and 24 meters long.
Correct Answer is :   8, 24
Q4The area of a rectangular book is 364 square centimeters. If its dimensions are a and a + 12, find the length and width of the book.
A. 16 cm, 28 cm
B. 14 cm, 26 cm
C. 15 cm, 27 cm
D. 13 cm, 25 cm

Solutions
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Step: 1
The area of a rectangle = length × width.
Step: 2
The area of the rectangular book = (a) × (a + 12) square centimeters.
Step: 3
364 = (a) × (a + 12)
  [Original equation.]
Step: 4
364 = a2 + 12a
  [Use distributive property.]
Step: 5
364 + 62 = a2 + 12a + 62
  [Add (122)2 = 62 = 36 to each side.]
Step: 6
400 = (a + 6)2
  [Write the right hand side as a perfect square and simplify.]
Step: 7
± 20 = (a + 6)
  [Evaluate square roots on both sides.]
Step: 8
± 20 - 6 = a + 6 - 6
  [Subtract 6 from each side.]
Step: 9
a = + 14 or - 26
  [Simplify.]
Step: 10
Width = a = 14 centimeters
  [Dimensions cannot be negative.]
Step: 11
Length = (a + 12) = (14 + 12) = 26 centimeters
  [Substitute 14 for a and add.]
Step: 12
The book is 14 centimeters wide and 26 centimeters long.
Correct Answer is :   14 cm, 26 cm
Q5A rectangular carpet measures a feet long and (a - 10) feet wide. What are the dimensions of the carpet, if its area is 56 square feet?
A. 15 feet by 5 feet
B. 13 feet by 3 feet
C. 14 feet by 4 feet
D. 16 feet by 6 feet

Solutions
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Step: 1
The area of a rectangle = Length × Width
Step: 2
The area of the rectangular carpet = (a) × (a - 10) square feet
Step: 3
56 = (a) × (a - 10)
  [Original equation.]
Step: 4
56 = a2 - 10a
  [Use distributive property.]
Step: 5
56 + (- 5)2 = a2 - 10a + (- 5)2
  [Add (- 102)2 = (- 5)2 = 25 to each side.]
Step: 6
81 = (a - 5)2
  [Write the right side of the equation as a perfect square and simplify.]
Step: 7
± 9 = (a - 5)
  [Evaluate square roots on both sides.]
Step: 8
± 9 + 5 = a - 5 + 5
  [Add 5 on each side.]
Step: 9
a = - 4 or 14
  [Simplify.]
Step: 10
Length of the rectangular carpet is a = 14 feet.
  [Dimensions cannot be negative.]
Step: 11
Width of the rectangular carpet is (a - 10) = (14 - 10) = 4 feet.
  [Repalce a with 14 and add.]
Step: 12
The dimensions of the carpet are 14 feet by 4 feet.
Correct Answer is :   14 feet by 4 feet
Q6Solve by graphing the related function of the equation 5x2 + 7 = 52.
A. 3
B. + 9 and - 9
C. - 3 and 3
D. 7

Solutions
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Step: 1
ax2 + bx + c = 0
  [The equation in standard form.]
Step: 2
5x2 + 7 = 52
  [Original equation.]
Step: 3
5x2 = 45
  [Subtract 7 from each side.]
Step: 4
x2 = 9
  [Divide with 5 on both sides.]
Step: 5
x2 - 9 = 0
  [Subtract 9 from each side.]
Step: 6
Sketch the graph of the related quadratic function y = x2 - 9
Step: 7
Estimate the values of the x-intercepts. From the graph, the x-intercepts appear to be - 3 and 3.
Step: 8
By substituting x = - 3 and x = 3 in x2 - 9 = 0, it can be observed that - 3 and 3 are solutions of the equation.
Correct Answer is :   - 3 and 3
Q7The graph of y = - 3x2 + 9x is shown in the figure. Estimate the solution of - 3x2 + 9x = 0 using the graph.


A. 0 and 1
B. 1 and 2
C. 0 and 2
D. 0 and 3

Solutions
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Step: 1
The x-intercepts in the graph of the equation y = ax2 + bx + c are the solutions of the related equation ax2 + bx + c = 0.
Step: 2
The graph intersects the x-axis at (0, 0) and (3, 0).
Step: 3
So, 0 and 3 are the solutions of the equation.
Correct Answer is :   0 and 3
Q8Find the solution of 2x2 - 8 = 0 using the graph.

A. 4 and - 4
B. 2 and - 2
C. 5 and - 5
D. 3 and - 3

Solutions
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Step: 1
2x2 - 8 = 0 is written in the standard form as y = 2x2 - 8.
Step: 2
The graph intersect the x-axis at points (- 2, 0) and (2, 0).
Step: 3
From the graph, the x-intercepts are - 2 and 2.
  [Estimate the values of the x-intercepts.]
Step: 4
2(- 2) 2 - 8 =0
  [Substitute x = - 2 in the equation 2x2 - 8 = 0.]
Step: 5
0 = 0
  [Simplify.]
Step: 6
2(2) 2 - 8 =0
  [Substitute x = 2 in the equation 2x2 - 8 =0.]
Step: 7
0 = 0
  [Simplify.]
Step: 8
Both the values satisfy the equation.
Step: 9
So, 2 and - 2 are the solutions of the equation.
Correct Answer is :   2 and - 2
Q9Use the graph to estimate the roots of the equation x2 + x = 2.


A. - 1 and 1
B. - 3 and 1
C. - 4 and 1
D. - 2 and 1

Solutions
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Step: 1
ax2 + bx + c = 0
  [The equation in standard form.]
Step: 2
x2 + x = 2
  [Original equation.]
Step: 3
x2 + x - 2 = 0
  [Subtract 2 from each side.]
Step: 4
Sketch the graph of the related quadratic equation y = x 2 + x - 2.
Step: 5
From the graph, x-intercepts are - 2 and 1.
Step: 6
(- 2) 2 + (- 2) - 2 = 0
  [Substitute x = - 2 in the equation.]
Step: 7
0 = 0
  [Simplify.]
Step: 8
(1)2 + 1 - 2 = 0
  [Substitute x = 1 in the equation.]
Step: 9
0 = 0
  [Simplify.]
Step: 10
The values x = - 2 and 1 satisfy the equation x2 + x - 2 = 0.
Step: 11
So, - 2 and 1 are the roots of the equation.
Correct Answer is :   - 2 and 1
Q10Estimate the solution of the equation 5x2 = 20 using a graph.


A. - 5 and 5
B. - 4 and 4
C. - 2 and 2
D. - 3 and 3

Solutions
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Step: 1
ax2 + bx + c = 0
  [The equation in standard form.]
Step: 2
5x2 = 20
  [Original equation.]
Step: 3
5x2 - 20 = 0
  [Subtract 20 from each side.]
Step: 4
x2 - 4 = 0
  [Divide by 5 on each side.]
Step: 5
Sketch the graph of the related quadratic function, y = x2 - 4.
Step: 6
From the graph, x-intercepts are - 2 and + 2.
Step: 7
(- 2) 2 - 4 = 0
  [Substitute x = - 2 in the equation.]
Step: 8
0 = 0
  [Simplify.]
Step: 9
(2) 2 - 4 = 0
  [Substitute x = 2 in the equation.]
Step: 10
0 = 0
  [Simplify.]
Step: 11
Both the values x = - 2 and x = 2 satisfy the equation x2 - 4 = 0.
Step: 12
So, the solutions of the equation are - 2 and 2.
Correct Answer is :   - 2 and 2
Q11Solve the quadratic equation - 4x2 - 4x + 8 = 0 using the graph.

A. - 3 and 1
B. - 4 and 1
C. - 2 and 1
D. - 2 and - 3

Solutions
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Step: 1
ax2 + bx + c = 0
  [The equation in standard form.]
Step: 2
- 4x2 - 4x + 8 = 0
  [Original equation.]
Step: 3
- x2 - x + 2 = 0
  [Divide each side by 4.]
Step: 4
Sketch the graph of the related quadratic function, y = - x2 - x + 2.
Step: 5
From the graph, x-intercepts are - 2 and 1.
Step: 6
- (- 2) 2 - (- 2) + 2 = 0
  [Substitute x = - 2 in the equation.]
Step: 7
0 = 0
  [Simplify.]
Step: 8
- (1) 2 - (1) + 2 = 0
  [Substitute x = 1 in the equation.]
Step: 9
0 = 0
  [Simplify.]
Step: 10
Both the values x = - 2 and x = 1 satisfy the equation.
Step: 11
So, the solutions of the equation are - 2 and 1.
Correct Answer is :   - 2 and 1
Q12Laura dives into a pool from the diving board, that is 16 feet high from the water. She dives with an initial downward velocity of - 24 feet per second. If the equation to model the height of the dive is h = - 16t2 + (- 24)t + 16, then find the time in seconds it takes Laura to reach the water.
A. 1
B. 1.5
C. 5.50
D. 0.50

Solutions
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Step: 1
h = - 16t2 + (- 24)t + 16
  [Original equation.]
Step: 2
0 = - 16t2 + (- 24t) + 16
  [Replace h with 0, as the height is zero at the water level.]
Step: 3
t = {-(-24)±[(-24)2-4(-16)(16)]}[2(-16)]
  [Substitute a = - 16, b = - 24 and c = 16 in the quadratic formula.]
Step: 4
t = 24±576+1024-32
  [Simplify.]
Step: 5
t = 24±1600-32
  [Simplify inside the radical.]
Step: 6
t = 24±40-32 = -2, 0.50
  [Simplify the radical.]
Step: 7
t = 0.50
  [Since t represents time, consider the positive integer.]
Correct Answer is :   0.50
Q13Jake jumped from a 729 feet bungee tower. Find the time it took him to reach the ground, if the equation that models his height is h = - 16t2 + 729, where t is the time in seconds.

A. 6.50
B. 7
C. 7.25
D. 6.75

Solutions
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Step: 1
h = - 16t2 + 729
  [Original equation.]
Step: 2
0 = - 16t2 + 729
  [Replace h with 0, as the height is zero at the ground level.]
Step: 3
t = {-(-0)±[(-0)2-4(-16)(729)]}[2(-16)]
  [Substitute the values in the quadratic formula: a = - 16, b = 0 and c = 729.]
Step: 4
= 0±(0+46656)-32
  [Simplify.]
Step: 5
= 0±46656-32
  [Simplify inside the radical.]
Step: 6
= 0±216-32
  [Simplify.]
Step: 7
= -216-32 = 6.75
  [Since t represents time, use the positive solution.]
Correct Answer is :   6.75
Q14The base of a triangle is 4 cm longer than the altitude. Find the length of the base if the area of the triangle is 48 cm2.
A. 12 cm
B. 8 cm
C. 14 cm
D. None of the above

Solutions
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Step: 1
Let x be the length of the altitude of the triangle & the length of its base = (x + 4) cm.
Step: 2
The area of a triangle = 12 × base × altitude
Step: 3
48 = 12 × (x + 4) × x
  [Substitute the values.]
Step: 4
48 = 12 × (x2 + 4x)
  [Distributive property.]
Step: 5
96 = (x2 + 4x)
  [Multiply throughout by 2.]
Step: 6
x2 + 4x - 96 = 0
  [Subtract 96 from the two sides of the equation.]
Step: 7
(x + 12)(x - 8) = 0
  [Factor.]
Step: 8
Therefore, x = - 12 or 8.
Step: 9
Reject the negative solution, as the length cannot be negative.
So, x = 8 and x + 4 = 8 + 4 = 12
Step: 10
So, the length of the base of the triangle is 12 cm.
Correct Answer is :   12 cm
Q15Solve by graphing the related function of the equation x2 + x = 2.

A. - 3 and 7
B. - 2 and 1
C. - 3 and 6
D. 2 and 1

Solutions
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Step: 1
x2 + x = 2
  [Original equation.]
Step: 2
x2 + x - 2 = 0
  [Subtract 2 from each side.]
Step: 3
Sketch the graph of the related quadratic function y = x2 + x - 2 as shown below.
Step: 4
From the graph, the x-intercepts appear to be - 2 and 1.
  [Estimate the values of the x-intercepts.]
Step: 5
So, - 2 and 1 are the solutions of the equation.
Correct Answer is :   - 2 and 1

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