Quadratic Equations: Master the Art of Solving Quadratics

Question 1

Which method is NOT used to solve quadratic equations?

Accepted Answer

Trial and error

Answer

Factorization

Answer

Completing the square

Answer

Using the quadratic formula

Question 2

The quadratic formula, used to solve quadratic equations of the form ax² + bx + c = 0, is:

Accepted Answer

x = (-b ± √(b² - 4ac)) / 2a

Answer

x = √(b² - 4ac)

Answer

x = b² - 4ac

Answer

x = (b² - 4ac) / a

Question 3

The discriminant of a quadratic equation ax² + bx + c = 0, which determines the nature of its roots, is:

Accepted Answer

b² - 4ac

Answer

a - b

Answer

a² + b²

Answer

a + b

Question 4

Which of the following is a characteristic of quadratic equations with complex roots?

Accepted Answer

They have no real roots.

Answer

They have three real roots.

Answer

They have one real root.

Answer

They have two real roots.

Question 5

Determine the equation of the parabola with vertex (2, -3) and focus (4, -3):

Accepted Answer

y = (x - 2)² - 3

Answer

y = (x + 2)² - 3

Answer

y = (x - 2)² + 3

Question 6

Which of the following is the equation of the axis of symmetry for the parabola y = x^2 + 4x + 3?

Accepted Answer

x = -2

Answer

y = 2

Answer

y = -2

Answer

x = 2

Question 7

Determine whether the parabola y = x^2 - 4x + 3 opens upward or downward:

Accepted Answer

Upward

Answer

Leftward

Answer

Rightward

Answer

Downward

Question 8

Find the discriminant of the quadratic equation: 2x^2 + 3x - 5 = 0.

Accepted Answer

17

Answer

5

Answer

-17

Answer

25

Question 9

Solve the quadratic inequality: x^2 - 3x + 2 > 0.

Accepted Answer

x < 1 or x > 2

Answer

x > 1 or x < 2

Answer

x < 1 and x > 2

Answer

x > 1 and x < 2

Question 10

Solve the quadratic equation x² - 5x + 6 = 0 by factorization:

Accepted Answer

x = 2 or x = 3

Answer

x = 1 or x = 3

Answer

x = 1 or x = 4

Answer

x = 2 or x = 4

Question 11

Find the vertex of the parabola represented by the equation y = x² - 4x + 3:

Accepted Answer

(2, -1)

Answer

(4, 1)

Answer

(1, -4)

Answer

(3, 2)

Question 12

Determine the domain of the function g(x) = √(x - 2):

Accepted Answer

[2, ∞)

Answer

(-∞, 2)

Answer

(-∞, ∞)

Answer

(2, ∞)

Question 13

Which of the following is NOT a method for solving quadratic equations?

Accepted Answer

Division

Answer

Completing the Square

Answer

Quadratic Formula

Answer

Factoring

Question 14

The graph of the quadratic function y = x^2 - 4x + 3 has:

Accepted Answer

A minimum at (2, -1)

Answer

A maximum at (2, -1)

Answer

A maximum at (2, 1)

Answer

A minimum at (2, 1)

Question 15

The equation of the axis of symmetry of the parabola y = 2(x - 1)^2 + 3 is:

Accepted Answer

x = 1

Answer

y = 2

Answer

y = 1

Answer

x = 2

Question 16

Which of the following is the standard form of a quadratic equation?

Accepted Answer

ax^2 + bx + c = 0, where a ≠ 0

Answer

ax + b = 0, where a ≠ 0

Answer

ax^2 + c = 0, where a ≠ 0

Answer

bx + c = 0, where b ≠ 0

Question 17

Determine the nature of the roots of the quadratic equation: x² - 8x + 16 = 0 using the discriminant

Accepted Answer

Real and equal

Answer

Complex

Answer

Real and distinct

Answer

Cannot be determined

Question 18

The quadratic formula is given by:

Accepted Answer

x = (-b ± √(b² - 4ac)) / 2a

Answer

x = (b ± √(b² - 4ac)) / 4a

Answer

x = (-b ± √(b² + 4ac)) / 2a

Answer

x = (b ± √(b² + 4ac)) / 4a

Question 19

The discriminant of a quadratic equation ax² + bx + c = 0 is:

Accepted Answer

b² - 4ac

Answer

a² - 4ac

Answer

4ac - b²

Answer

-b² + 4ac

Question 20

When the discriminant of a quadratic equation is positive, the equation has:

Accepted Answer

Two distinct real solutions

Answer

Two complex solutions

Answer

Zero solutions

Question 21

The equation of the axis of symmetry of the parabola y = x² + 2x - 3 is:

Accepted Answer

x = -1

Answer

y = 1

Answer

y = -1

Answer

x = 1

Question 22

Factorise the following quadratic expression: x^2 - 6x + 8

Accepted Answer

(x - 2)(x - 4)

Answer

(x - 2)(x + 4)

Answer

(x + 2)(x + 4)

Question 23

Solve the following quadratic equation by completing the square: x^2 + 4x - 5 = 0

Accepted Answer

x = -2 ± √7

Answer

x = 2 ± √7

Answer

x = -2 ± i√7

Answer

x = 2 ± i√7

Question 24

Use the quadratic formula to solve the equation: 2x^2 - 5x + 2 = 0

Accepted Answer

x = (5 ± i√7)/4

Answer

x = (-5 ± i√7)/4

Answer

x = (-5 ± √7)/4

Answer

x = (5 ± √7)/4

Question 25

Find the discriminant of the quadratic equation: ax^2 + bx + c = 0

Accepted Answer

b^2 - 4ac

Answer

-b^2 + 4ac

Answer

b^2 + 4ac

Answer

4ac - b^2

Question 26

What is the nature of the solutions to the quadratic equation: x^2 + 2x + 1 = 0?

Accepted Answer

Real and equal

Answer

Imaginary

Answer

Complex and conjugate

Answer

Real and distinct

Question 27

Solve the following quadratic equation using any method: 3x^2 - 2x - 5 = 0

Accepted Answer

x = (2 ± i√7)/3

Answer

x = (-2 ± √7)/3

Answer

x = (2 ± √7)/3

Answer

x = (-2 ± i√7)/3

Question 28

Find the vertex of the parabola represented by the equation: y = -x^2 + 4x - 3

Accepted Answer

(2, 1)

Answer

(4, 2)

Answer

(1, 2)

Answer

(-2, 1)

Question 29

Which of the following is the equation of the axis of symmetry of the parabola: y = x^2 - 2x + 1?

Accepted Answer

x = 1

Answer

y = 2

Answer

y = 1

Answer

x = 2

Question 30

Find the range of values of k for which the quadratic equation: x^2 + kx + 2 = 0 has two distinct real solutions.

Accepted Answer

k < -4

Answer

k <-4

Answer

k >4

Answer

k > -4

Question 31

A ball is thrown vertically upwards with a velocity of 15 m/s. The height of the ball at time t seconds is given by the equation: h(t) = -5t^2 + 15t. What is the maximum height reached by the ball?

Accepted Answer

11.25 m

Answer

15 m

Answer

7.5 m

Answer

22.5 m

Question 32

Which of the following is a quadratic equation?

Accepted Answer

x² + 3x - 2 = 0

Answer

x³ - 2x² + 1 = 0

Answer

2x + 1 = 3x - 4

Answer

x + 5 = 0

Question 33

What is the discriminant of the quadratic equation 2x² - 5x + 3 = 0?

Accepted Answer

1

Answer

-1

Answer

-13

Answer

13

Question 34

What does the discriminant tell us about the nature of the solutions to a quadratic equation?

Accepted Answer

The number and type of solutions

Answer

The sum of the solutions

Answer

The coefficient of the linear term

Answer

The product of the solutions

Question 35

Solve the quadratic equation x² - 6x + 8 = 0 by factorization.

Accepted Answer

x = 2 or x = 4

Answer

x = -2 or x = -4

Answer

x = -1 or x = -8

Answer

x = 1 or x = 8

Question 36

Solve the quadratic equation x² + 4x - 5 = 0 by completing the square.

Accepted Answer

x = 1 or x = -5

Answer

x = -1 or x = 5

Answer

x = 2 or x = -2.5

Answer

x = -2 or x = 2.5

Question 37

What are the solutions to the quadratic equation 3x² - 2x - 1 = 0 using the quadratic formula?

Accepted Answer

x = 1 or x = -1/3

Answer

x = 2 or x = -1/2

Answer

x = -2 or x = 1/2

Answer

x = -1 or x = 1/3

Question 38

A rectangular garden has a length that is 3 meters longer than its width. If the area of the garden is 28 square meters, what is the width of the garden?

Accepted Answer

4 meters

Answer

11 meters

Answer

3 meters

Answer

7 meters

Question 39

The height of a ball thrown vertically upwards is given by the equation h = -5t² + 20t, 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

20 meters

Answer

10 meters

Answer

40 meters

Answer

5 meters

Question 40

Which of the following quadratic equations has two distinct real roots?

Accepted Answer

x² - 4x + 3 = 0

Answer

x² + 2x + 1 = 0

Answer

x² + 4x + 4 = 0

Answer

x² - 2x + 1 = 0

Question 41

The graph of a quadratic function is a parabola. What is the relationship between the discriminant of the quadratic equation and the number of x-intercepts of the parabola?

Accepted Answer

A positive discriminant corresponds to two x-intercepts.

Answer

A zero discriminant corresponds to one x-intercept.

Answer

A negative discriminant corresponds to two x-intercepts.

Question 42

What is the discriminant of the quadratic equation $3x^2 - 2x - 1 = 0$?

Accepted Answer

16

Answer

12

Answer

4

Answer

-8

Question 43

A projectile's height (h) in meters above the ground after t seconds is given by the equation  $h = -5t^2 + 20t + 10$. What is the maximum height the projectile reaches?

Accepted Answer

30 meters

Answer

20 meters

Answer

10 meters

Answer

40 meters

Question 44

Which of the following is a factorization of the quadratic expression $2x^2 + 5x - 3$ ?

Accepted Answer

(2x - 1)(x + 3)

Answer

(x - 1)(2x + 3)

Answer

(x + 1)(2x - 3)

Answer

(2x + 1)(x - 3)

Question 45

The solutions of the quadratic equation $x^2 - 6x + k = 0$ are equal. What is the value of k?

Accepted Answer

9

Answer

3

Answer

6

Question 46

What is the sum of the roots of the quadratic equation $x^2 - 7x + 10 = 0$?

Accepted Answer

7

Answer

10

Answer

-10

Answer

-7

Question 47

Solve the equation $x^2 + 4x - 12 = 0$ by completing the square.

Accepted Answer

x = 2 or x = -6

Answer

x = -3 or x = 4

Answer

x = 3 or x = -4

Answer

x = -2 or x = 6

Question 48

A rectangle's length is 3 cm more than twice its width. If the area of the rectangle is 27 cm², what is its width?

Accepted Answer

3 cm

Answer

12 cm

Answer

9 cm

Answer

6 cm

Question 49

The graph of the quadratic function $y = ax^2 + bx + c$ opens downwards. Which of the following statements about 'a' is true?

Accepted Answer

a < 0

Answer

a > 0

Answer

a = 0

Question 50

Solve for x: $2x^2 - 7x + 3 = 0$ using the quadratic formula.

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 51

The discriminant of a quadratic equation is negative. What does this tell you about its solutions?

Accepted Answer

The equation has no real solutions.

Answer

The equation has infinitely many solutions.

Answer

The equation has two distinct real solutions.

Answer

The equation has one real solution (a repeated root).