Quadratic Equations: Master the Art of Solving Quadratics

Question 1

Solve the quadratic equation x^2 + 6x + 5 = 0 by completing the square.

Accepted Answer

x = -1 ± 2i

Answer

x = -2 ± i

Answer

x = -2 ± 2i

Answer

x = -1 ± i

Question 2

Sketch the graph of the quadratic function y = -x² + 4x - 3.

Accepted Answer

Parabola opening downwards, vertex at (2, 1)

Answer

Parabola opening upwards, vertex at (2, 1)

Answer

Parabola opening downwards, vertex at (-2, 1)

Answer

Parabola opening upwards, vertex at (-2, 1)

Question 3

Which of the following methods is NOT used to solve quadratic equations?

Accepted Answer

Using a graphing calculator

Answer

Factoring

Answer

Completing the square

Answer

Using the quadratic formula

Question 4

Factor the quadratic expression x^2 - 5x + 6

Accepted Answer

(x - 2)(x - 3)

Answer

(x - 6)(x + 1)

Answer

(x + 2)(x - 3)

Answer

(x + 6)(x - 1)

Question 5

Which method is NOT used to solve quadratic equations?

Accepted Answer

Substitution

Answer

Factoring

Answer

Completing the Square

Answer

Quadratic Formula

Question 6

Which of the following is the equation of the axis of symmetry of the graph of the quadratic function y = ax^2 + bx + c?

Accepted Answer

x = -b / 2a

Answer

x = b / 2a

Answer

y = -b / 2a

Answer

y = b / 2a

Question 7

Use the quadratic formula to find the discriminant of the equation 2x^2 + 5x - 3 = 0.

Accepted Answer

25

Answer

9

Answer

-25

Answer

15

Question 8

Which of the following transformations will shift the graph of the parabola represented by y = x^2 to the left by 3 units?

Accepted Answer

y = (x - 3)^2

Answer

y = (x + 3)^2

Answer

y = x^2 + 3

Answer

y = x^2 - 3

Question 9

Which of the following represents the real solutions to the equation x² - 4 = 0?

Accepted Answer

±2

Answer

2

Answer

-2

Answer

There are no real solutions

Question 10

Determine the roots of the quadratic equation x² - 5x + 6 = 0.

Accepted Answer

x = -2 and x = -3

Answer

x = 6 and x = 1

Answer

x = 3 and x = 2

Answer

x = -3 and x = -2

Question 11

Which of the following is the graph of the quadratic function f(x) = x² - 4x + 3?

Accepted Answer

A parabola opening upwards with vertex (2, -1)

Answer

A parabola opening downwards with vertex (2, -1)

Answer

A parabola opening upwards with vertex (2, 1)

Answer

A parabola opening downwards with vertex (2, 1)

Question 12

Find the maximum or minimum value of the quadratic function f(x) = x² - 8x + 15.

Accepted Answer

Minimum value: 3

Answer

Maximum value: 3

Answer

Maximum value: -3

Answer

Minimum value: -3

Question 13

Which of the following is a quadratic equation?

Accepted Answer

2x² + 5x - 3 = 0

Answer

x³ - 2x² + 1 = 0

Answer

x + 3 = 0

Answer

2x + 5 = 0

Question 14

Solve the equation 2x² - 5x - 3 = 0 by factoring.

Accepted Answer

x = 3 or x = -1/2

Answer

x = 1 or x = 3/2

Answer

x = -3 or x = 1/2

Answer

x = -1 or x = -3/2

Question 15

What is the value of the discriminant of the quadratic equation x² + 6x + 9 = 0?

Accepted Answer

0

Answer

-12

Answer

36

Answer

12

Question 16

Solve the equation x² - 4x + 2 = 0 using the quadratic formula.

Accepted Answer

x = 2 ± √2

Answer

x = -2 ± 2√2

Answer

x = -2 ± √2

Answer

x = 2 ± 2√2

Question 17

What is the vertex of the parabola represented by the equation y = x² - 4x + 3?

Accepted Answer

(2, -1)

Answer

(3, 0)

Answer

(-2, 1)

Answer

(1, 0)

Question 18

The roots of a quadratic equation are 3 and -2. Which of the following is the equation?

Accepted Answer

x² - x - 6 = 0

Answer

x² - x + 6 = 0

Answer

x² + x + 6 = 0

Answer

x² + x - 6 = 0

Question 19

A rectangular garden has a length that is 5 meters more than its width. If the area of the garden is 150 square meters, what is the width of the garden?

Accepted Answer

10 meters

Answer

5 meters

Answer

15 meters

Answer

20 meters

Question 20

The graph of the quadratic function y = -x² + 2x + 3 intersects the x-axis at which points?

Accepted Answer

(-1, 0) and (3, 0)

Answer

(-1, 0) and (-3, 0)

Answer

(1, 0) and (3, 0)

Answer

(1, 0) and (-3, 0)

Question 21

What is the axis of symmetry of the parabola represented by the equation y = 2x² - 8x + 5?

Accepted Answer

x = 2

Answer

x = 4

Answer

x = -4

Answer

x = -2

Question 22

For what value of 'k' does the equation kx² - 4x + 1 = 0 have exactly one solution?

Accepted Answer

4

Answer

-4

Answer

1

Answer

0

Question 23

What is the discriminant of the quadratic equation $3x^2 - 2x - 5 = 0$?

Accepted Answer

64

Answer

4

Answer

-64

Answer

16

Question 24

The graph of a quadratic function intersects the x-axis at x = -2 and x = 4. What is the equation of the axis of symmetry?

Accepted Answer

x = 1

Answer

x = 2

Answer

x = -2

Question 25

Which of the following quadratic equations has no real roots?

Accepted Answer

$x^2 + 2x + 5 = 0$

Answer

$2x^2 + x - 1 = 0$

Answer

$x^2 - 2x - 3 = 0$

Answer

$-x^2 + 3x + 2 = 0$

Question 26

A rectangle has a length that is 3 cm longer than its width. If the area of the rectangle is 50 cm², what is the width?

Accepted Answer

5 cm

Answer

12 cm

Answer

10 cm

Answer

7 cm

Question 27

What are the roots of the quadratic equation $2x^2 - 7x - 4 = 0$, using the quadratic formula?

Accepted Answer

$x = \frac{7 \pm \sqrt{81}}{4}$

Answer

$x = \frac{-7 \pm \sqrt{17}}{4}$

Answer

$x = \frac{-7 \pm \sqrt{81}}{2}$

Answer

$x = \frac{7 \pm \sqrt{17}}{2}$

Question 28

Which of the following is the vertex form of the quadratic equation $y = x^2 - 6x + 5$?

Accepted Answer

$y = (x - 3)^2 - 4$

Answer

$y = (x + 3)^2 + 4$

Answer

$y = (x + 3)^2 - 4$

Answer

$y = (x - 3)^2 + 4$

Question 29

The height of a ball thrown vertically upwards is given by the equation $h(t) = -5t^2 + 20t + 2$, where h is the height in meters and t is the time in seconds. What is the maximum height reached by the ball?

Accepted Answer

22 meters

Answer

5 meters

Answer

2 meters

Question 30

For what values of 'k' does the equation $x^2 + 4x + k = 0$ have equal roots?

Accepted Answer

k = 4

Answer

k = 2

Answer

k = -4

Answer

k = -2

Question 31

Which of the following is a factor of the quadratic expression $6x^2 - 7x - 3$?

Accepted Answer

$(2x - 3)$

Answer

$(3x - 1)$

Answer

$(3x + 3)$

Answer

$(2x + 1)$

Question 32

The graph of a quadratic function is a parabola that opens downwards and has a y-intercept of (0, 6). Which of the following could be the equation of the function?

Accepted Answer

$y = -2x^2 + 6$

Answer

$y = -2(x - 6)^2$

Answer

$y = 2(x + 6)^2$

Answer

$y = 2x^2 + 6$

Question 33

What is the solution to the quadratic equation 2x² - 5x - 3 = 0?

Accepted Answer

x = 3 or x = -1/2

Answer

x = -3 or x = 1/2

Answer

x = 2 or x = -3/2

Answer

x = -2 or x = 3/2

Question 34

The graph of the quadratic function y = x² + 4x + 3 intersects the x-axis at which points?

Accepted Answer

(-3, 0) and (-1, 0)

Answer

(1, 0) and (-3, 0)

Answer

(3, 0) and (1, 0)

Answer

(3, 0) and (-1, 0)

Question 35

Which of the following is the correct formula for the discriminant of a quadratic equation ax² + bx + c = 0?

Accepted Answer

b² - 4ac

Answer

a² + 4bc

Answer

a² - 4bc

Answer

c² - 4ab

Question 36

Solve the equation 3x² - 6x + 2 = 0 using the quadratic formula.

Accepted Answer

x = 1 ± √(1/3)

Answer

x = -1 ± √(7/3)

Answer

x = -1 ± √(1/3)

Answer

x = 1 ± √(7/3)

Question 37

A rectangular garden has a length that is 3 meters more than its width. If the area of the garden is 54 square meters, what is the width of the garden?

Accepted Answer

6 meters

Answer

9 meters

Answer

12 meters

Answer

3 meters

Question 38

Complete the square for the expression x² - 8x + ______

Accepted Answer

16

Answer

8

Answer

4

Answer

64

Question 39

The equation x² + 2x - 8 = 0 has how many real roots?

Accepted Answer

Two

Answer

None

Answer

Infinitely many

Answer

One

Question 40

What is the vertex of the parabola represented by the equation y = -x² + 4x - 3?

Accepted Answer

(2, 1)

Answer

(-2, 1)

Answer

(1, 2)

Question 41

The equation (x - 2)² = 9 is equivalent to which of the following?

Accepted Answer

x² - 4x - 5 = 0

Answer

x² + 4x - 5 = 0

Answer

x² + 4x + 5 = 0

Answer

x² - 4x + 5 = 0

Question 42

Which of the following describes the graph of the quadratic function y = 2x² - 8x + 6?

Accepted Answer

A parabola opening upwards with vertex at (2, -2)

Answer

A parabola opening downwards with vertex at (-2, 2)

Answer

A parabola opening downwards with vertex at (2, -2)

Answer

A parabola opening upwards with vertex at (-2, 2)