Master Quadratic Equations: Online Quizzes and Practice

Question 1

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

Accepted Answer

(x - 2)(x - 3) = 0

Answer

(x + 2)(x + 3) = 0

Answer

(x + 2)(x - 3) = 0

Answer

(x - 2)(x + 3) = 0

Question 2

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

Accepted Answer

(2, -1)

Answer

(0, 3)

Answer

(4, -1)

Answer

(-2, -1)

Question 3

Identify the type of conic section represented by the equation: 4x² + 9y² - 36x + 54y - 31 = 0

Accepted Answer

Ellipse

Answer

Hyperbola

Answer

Circle

Answer

Parabola

Question 4

Find the roots of the equation: $$x^2 + 2x - 8 = 0$$

Accepted Answer

x = -4 or x = 2

Answer

x = 2 or x = 4

Answer

x = 1 or x = -8

Answer

x = -2 or x = 8

Question 5

Solve the quadratic inequality: x² - 5x + 6 < 0

Accepted Answer

2 < x < 3

Answer

x > 2 or x < 3

Answer

x < 2 or x > 3

Answer

x > 0 and x < 6

Question 6

Which of the following is the graph of the quadratic equation: y = x² - 4x + 4?

Accepted Answer

(Parabola opening upwards with vertex at (2, 0))

Answer

(Circle)

Answer

(Parabola opening downwards with vertex at (2, 0))

Answer

(Line with a positive slope)

Question 7

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

Accepted Answer

29

Answer

13

Answer

-13

Question 8

Solve the equation: (x - 3)^2 = 16.

Accepted Answer

x = 7 or x = -1

Answer

x = 4 or x = -8

Answer

x = 8 or x = -4

Question 9

Determine the number of real solutions for the equation: x^2 - 2x + 2 = 0.

Accepted Answer

0

Answer

2

Answer

1

Answer

Cannot be determined

Question 10

Solve the inequality: x^2 - 5x + 6 > 0.

Accepted Answer

x < 2 or x > 3

Answer

x > -2 or x < -3

Answer

x < -2 or x > -3

Answer

x < 3 or x > 2

Question 11

Identify the type of conic section represented by the equation: 4x^2 + 9y^2 - 36x + 54y - 81 = 0.

Accepted Answer

Ellipse

Answer

Parabola

Answer

Hyperbola

Answer

Circle

Question 12

Which of the following represents the graph of a parabola that opens downwards?

Accepted Answer

y = -x^2

Answer

y = x^2

Answer

y = 1/x^2

Answer

y = x^3

Question 13

Find the vertex of the parabola: y = (x - 2)^2 + 1

Accepted Answer

(2, 1)

Answer

(4, 1)

Answer

(-2, 1)

Answer

(1, 2)

Question 14

Determine whether the following equation is a quadratic equation: 3x^3 + 2x^2 - 5x + 1 = 0

Accepted Answer

No, it is a cubic equation.

Answer

Maybe

Answer

Yes

Answer

Not sure

Question 15

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

Accepted Answer

ax^2 + bx + c = 0, where a ≠ 0

Answer

r = a(1 - cosθ)

Answer

x^2 + y^2 = z^2

Answer

y = mx + b

Question 16

Solve for x: (x + 3)(x - 5) = 0

Accepted Answer

x = -3 or x = 5

Answer

x = 2 or x = 8

Answer

x = -2 or x = -8

Answer

x = 3 or x = -5

Question 17

Factor the quadratic expression **x^2 - 9**:

Accepted Answer

**(x + 3)(x - 3)**

Answer

**Incorrect: (x - 3)(x + 9)**

Answer

**Incorrect: (x + 9)(x + 1)**

Question 18

Solve the quadratic equation by completing the square: **x^2 + 6x - 7 = 0**

Accepted Answer

**x = -3 ± 2√2**

Answer

**Incorrect: x = 3 ± √7**

Answer

**Incorrect: x = -3 ± √2**

Answer

**Incorrect: x = 3 ± 2√7**

Question 19

Use the quadratic formula to solve the equation: **2x^2 - 5x + 3 = 0**

Accepted Answer

**x = (5 ± √13) / 4**

Answer

**Incorrect: x = (5 ± √11) / 2**

Answer

**Incorrect: x = (-5 ± √13) / 4**

Question 20

Determine the discriminant of the quadratic equation: **x^2 - 4x + c = 0**

Accepted Answer

**16 - 4c**

Answer

**Incorrect: 4 - 16c**

Answer

**Incorrect: 16c - 4**

Answer

**Incorrect: c - 16**

Question 21

Find the vertex of the parabola represented by the equation: **y = x^2 - 2x + 5**

Accepted Answer

**Vertex = (1, 4)**

Answer

**Incorrect: (-1, 6)**

Answer

**Incorrect: (2, 3)**

Answer

**Incorrect: (0, 5)**

Question 22

Which of the following is the graph of the quadratic equation: **y = -x^2 + 4x - 3**?

Accepted Answer

**A parabola opening downwards**

Answer

**Incorrect: A straight line**

Answer

**Incorrect: A parabola opening upwards**

Answer

**Incorrect: A circle**

Question 23

Solve the quadratic inequality: **x^2 - 5x + 6 > 0**

Accepted Answer

**x < 2 or x > 3**

Answer

**Incorrect: x < 2 or x < 3**

Answer

**Incorrect: x > 2 or x < 3**

Answer

**Incorrect: x < 3 or x > 2**

Question 24

Which of the following is the equation of the axis of symmetry for the parabola: **y = (x - 2)^2 + 3**?

Accepted Answer

**x = 2**

Answer

**Incorrect: y = 2**

Answer

**Incorrect: y = 3**

Answer

**Incorrect: x = 3**

Question 25

Find the domain and range of the quadratic function: **f(x) = -x^2 + 2x + 1**

Accepted Answer

**Domain: all real numbers; Range: (-∞, 1]**

Answer

**Incorrect: Domain: [0, 2]; Range: (-∞, 1]**

Answer

**Incorrect: Domain: all real numbers; Range: [0, ∞)**

Question 26

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

Accepted Answer

(x - 2)(x - 3) = 0

Answer

(x + 2)(x + 3) = 0

Answer

(x - 1)(x - 6) = 0

Question 27

Given the discriminant of a quadratic equation is 16, what type of solutions does it have?

Accepted Answer

Two distinct real solutions

Answer

Two complex solutions

Answer

No real solutions

Answer

One real solution

Question 28

Determine the equation of the axis of symmetry for the parabola: y = (x - 3)² + 2

Accepted Answer

x = 3

Answer

y = 2

Answer

y = 3

Answer

x = 2

Question 29

Find the range of the function: f(x) = 2x² - 4x + 3

Accepted Answer

[1, ∞)

Answer

[1, 3]

Answer

(-∞, 1]

Answer

(-∞, ∞)

Question 30

Identify the quadratic inequality from the following:

Accepted Answer

x² - 4x + 4 < 0

Answer

x - 4 < 0

Answer

x² + 4x + 4 < 0

Answer

x³ - 4x + 4 < 0

Question 31

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

Accepted Answer

(x - 2)(x - 3)

Answer

(x - 1)(x - 6)

Answer

(x + 2)(x - 3)

Answer

(x + 1)(x - 5)

Question 32

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

Accepted Answer

ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0

Answer

ax³ + bx² + cx + d = 0, where a, b, c, and d are real numbers

Answer

ax + b = 0, where a and b are real numbers and a ≠ 0

Answer

ax + by + c = 0, where a, b, and c are real numbers and a, b ≠ 0

Question 33

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

Accepted Answer

Linear factorization

Answer

Quadratic formula

Answer

Factoring

Answer

Completing the square

Question 34

Solve the quadratic equation x² - 5x + 6 = 0 by factoring.

Accepted Answer

x = 2, x = 3

Answer

x = -1, x = -6

Answer

x = 1, x = 6

Answer

x = -2, x = -3

Question 35

Complete the square to solve the equation x² + 6x + 5 = 0.

Accepted Answer

x = -1, x = -5

Answer

x = -3, x = -2

Answer

x = 3, x = 2

Answer

x = 1, x = 5

Question 36

The discriminant of a quadratic equation determines the nature of its roots. What does a discriminant of 0 indicate?

Accepted Answer

One real, repeated root

Answer

Two distinct real roots

Answer

No real roots

Answer

Two complex roots

Question 37

A ball is thrown vertically upwards with an initial velocity of 20 m/s. The height (h) of the ball after t seconds is given by the equation h = -5t² + 20t. At what time does the ball reach its maximum height?

Accepted Answer

t = 2 seconds

Answer

t = 1 second

Answer

t = 3 seconds

Question 38

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

Accepted Answer

(2, -1)

Answer

(-2, 11)

Answer

(1, 0)

Answer

(0, 3)

Question 39

The graph of a quadratic equation is a parabola. What is the shape of the parabola when the coefficient of the x² term is negative?

Accepted Answer

Opens downwards

Answer

Circular

Answer

Linear

Answer

Opens upwards

Question 40

Factor the quadratic trinomial: x² - 5x + 6

Accepted Answer

(x - 2)(x - 3) = 0

Answer

(x - 2)(x + 3) = 0

Answer

(x + 2)(x + 3) = 0

Question 41

Solve the quadratic equation using the quadratic formula: 2x² + 5x - 3 = 0

Accepted Answer

x = (-5 ± √29) / 4

Answer

x = (5 ± √29) / 4

Answer

x = (-5 ± √19) / 4

Answer

x = (5 ± √19) / 4

Question 42

Determine the discriminant of a general quadratic equation: ax² + bx + c = 0.

Accepted Answer

b² - 4ac

Answer

a² - 4bc

Answer

a² + 2bc + c²

Answer

4a² - b² + c²

Question 43

Find the equation of a parabola with the vertex at (h, k) and passing through the point (x, y).

Accepted Answer

y = a(x - h)² + k, where a = (y - k) / (x - h)²

Answer

y = (x + h)² - k, where a = (y + k) / (x + h)²

Answer

y = (x - h)² + k, where a = (x - h) / (y - k)²

Answer

y = (x - h)² - k, where a = (y + k) / (x - h)²

Question 44

Find the number of real solutions to the quadratic equation: ax² + bx + c = 0, where the discriminant is negative.

Accepted Answer

No real solutions

Answer

Two real solutions

Answer

One real solution

Answer

Infinitely many real solutions

Question 45

Find the sum of the roots of the quadratic equation: ax² + bx + c = 0.

Accepted Answer

-b / a

Answer

-c / b

Answer

c / b

Answer

b / a

Question 46

Solve the quadratic equation: ax² + bx + c = 0, where a, b, and c are real numbers.

Accepted Answer

x = (-b ± √(b² - 4ac)) / 2a

Answer

x = (±b ± √(4ac + b²)) / 2a

Answer

x = (-b ± √(4ac - b²)) / 2a

Answer

x = (±b ± √(b² - 4ac)) / 2a

Question 47

Which of the following expressions is the factored form of the quadratic expression: ax² + bx + c?

Accepted Answer

(x - r)(x - s), where r and s are the roots of the quadratic equation.

Answer

(x - r)(x + s), where r and s are the roots of the quadratic equation.

Answer

(x + r)(x + s), where r and s are the roots of the quadratic equation.

Answer

(x + r)(x - s), where r and s are the roots of the quadratic equation.

Question 48

To complete the square of the quadratic expression x² + bx, what number should be added to both sides of the equation?

Accepted Answer

b²/4

Answer

b/2

Answer

-b²/4

Answer

b²/8

Question 49

In the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a, which variable represents the constant term of the quadratic equation?

Accepted Answer

c

Answer

a

Answer

d

Answer

b

Question 50

Find the equation of the parabola with the given vertex and a y-intercept of (0, d).

Accepted Answer

y = a(x - h)² + k, where a = (d / h²), h is the x-coordinate of the vertex, and k is the y-coordinate of the vertex.

Answer

y = a(x + h)² + k, where a = (d / h²), h is the x-coordinate of the vertex, and k is the y-coordinate of the vertex.

Answer

y = a(x - h)² + k, where a = (h² / d), h is the x-coordinate of the vertex, and k is the y-coordinate of the vertex.

Answer

y = a(x + h)² + k, where a = (h² / d), h is the x-coordinate of the vertex, and k is the y-coordinate of the vertex.

Question 51

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

Accepted Answer

ax² + bx + c = 0, where the discriminant is positive.

Answer

ax² + bx + c = 0, where the discriminant is negative.

Answer

ax² + bx + c = 0, where the coefficient of x² is zero.

Answer

ax² + bx + c = 0, where the discriminant is zero.

Question 52

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

Accepted Answer

2.5

Answer

-2.5

Answer

-1.5

Answer

1.5

Question 53

Which of the following is a quadratic equation?

Accepted Answer

y = 3x² - 2x + 1

Answer

y = 2x + 5

Answer

y = 1/x

Answer

y = x³ - 4x + 2