Step: 1

From the figure, the graph intersects the x - axis at two points (-1, 0) and ( 3 2 , 0).

Step: 2

The x -intercepts of the curve are -1 and 3 2 .

Step: 3

2x ^{2} - x = 3

[Original equation]

Step: 4

2(-1)^{2} - (-1) = 3

[Substitute x -intercept as -1.]

Step: 5

3 = 3

[Simplify.]

Step: 6

2( 3 2 )^{2} - (3 2 ) = 3

[Substitute x -intercept as 3 2 .]

Step: 7

3 = 3

[Simplify.]

Step: 8

Both the values satisfy the equation. So, -1 and 3 2 are the solutions of the equation.

Correct Answer is : -1 and 3 2

Step: 1

- 3x ^{2} = - 27

[Original equation]

Step: 2

[Divide - 3 on each side]

Step: 3

[Subtract 9 from each side]

Step: 4

[Simplify.]

Step: 5

Sketch the graph of the related quadratic function y = x ^{2} - 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

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 = a ^{2} + 16a

[Use distributive property.]

Step: 5

192 + 8^{2} = a ^{2} + 16a + 8^{2}

[Add (16 2 )^{2} = 8^{2} = 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

[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

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 = a ^{2} + 12a

[Use distributive property.]

Step: 5

364 + 6^{2} = a ^{2} + 12a + 6^{2}

[Add (12 2 )^{2} = 6^{2} = 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

[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

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 = a ^{2} - 10a

[Use distributive property.]

Step: 5

56 + (- 5)^{2} = a ^{2} - 10a + (- 5)^{2}

[Add (- 10 2 )^{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

[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

Step: 1

[The equation in standard form.]

Step: 2

5x ^{2} + 7 = 52

[Original equation.]

Step: 3

5x ^{2} = 45

[Subtract 7 from each side.]

Step: 4

[Divide with 5 on both sides.]

Step: 5

[Subtract 9 from each side.]

Step: 6

Sketch the graph of the related quadratic function y = x ^{2} - 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 x ^{2} - 9 = 0, it can be observed that - 3 and 3 are solutions of the equation.

Correct Answer is : - 3 and 3

Step: 1

The x -intercepts in the graph of the equation y = a x ^{2} + b x + c are the solutions of the related equation a x ^{2} + b x + 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

Step: 1

2x ^{2} - 8 = 0 is written in the standard form as y = 2x ^{2} - 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 2x ^{2} - 8 = 0.]

Step: 5

0 = 0

[Simplify.]

Step: 6

2(2) ^{2} - 8 =0

[Substitute x = 2 in the equation 2x ^{2} - 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

Step: 1

[The equation in standard form.]

Step: 2

[Original equation.]

Step: 3

[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 x ^{2} + x - 2 = 0.

Step: 11

So, - 2 and 1 are the roots of the equation.

Correct Answer is : - 2 and 1

Step: 1

[The equation in standard form.]

Step: 2

5x ^{2} = 20

[Original equation.]

Step: 3

5x ^{2} - 20 = 0

[Subtract 20 from each side.]

Step: 4

[Divide by 5 on each side.]

Step: 5

Sketch the graph of the related quadratic function, y = x ^{2} - 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 x ^{2} - 4 = 0.

Step: 12

So, the solutions of the equation are - 2 and 2.

Correct Answer is : - 2 and 2

Step: 1

[The equation in standard form.]

Step: 2

- 4x ^{2} - 4x + 8 = 0

[Original equation.]

Step: 3

- x ^{2} - x + 2 = 0

[Divide each side by 4.]

Step: 4

Sketch the graph of the related quadratic function, 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

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

Step: 1

[Original equation.]

Step: 2

0 = - 16t ^{2} + (- 24t ) + 16

[Replace h with 0, as the height is zero at the water level.]

Step: 3

[Substitute a = - 16, b = - 24 and c = 16 in the quadratic formula.]

Step: 4

[Simplify.]

Step: 5

[Simplify inside the radical.]

Step: 6

[Simplify the radical.]

Step: 7

[Since t represents time, consider the positive integer.]

Correct Answer is : 0.50

Step: 1

[Original equation.]

Step: 2

0 = - 16t ^{2} + 729

[Replace h with 0, as the height is zero at the ground level.]

Step: 3

[Substitute the values in the quadratic formula: a = - 16, b = 0 and c = 729.]

Step: 4

= 0 ± ( 0 + 4 6 6 5 6 ) - 3 2

[Simplify.]

Step: 5

= 0 ± 4 6 6 5 6 - 3 2

[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

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 = 1 2 × base × altitude

Step: 3

48 = 1 2 × (x + 4) × x

[Substitute the values.]

Step: 4

48 = 1 2 × (x ^{2} + 4x )

[Distributive property.]

Step: 5

96 = (x ^{2} + 4x )

[Multiply throughout by 2.]

Step: 6

[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

So,

Step: 10

So, the length of the base of the triangle is 12 cm.

Correct Answer is : 12 cm

Step: 1

[Original equation.]

Step: 2

[Subtract 2 from each side.]

Step: 3

Sketch the graph of the related quadratic function y = x ^{2} + 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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