Algebra I

Solving Quadratic Equations

Lesson

A quadratic equation has an $x^2$ term and is set equal to zero:

ax^2 + bx + c = 0

It usually has two solutions. There are two methods to know.

Method 1: Solve by factoring

Factor the polynomial, then use the zero-product property: if a product equals zero, at least one factor must be zero.

Worked example 1

x^2 + 5x + 6 = 0

Factor:

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

Set each factor to zero:

x + 2 = 0 \quad \text{or} \quad x + 3 = 0

x = -2 \quad \text{or} \quad x = -3

Method 2: Solve by square roots

When the equation has the form $x^2 = k$ , take the square root of both sides. Don’t forget the ± — both the positive and negative roots are solutions.

Worked example 2

x^2 = 49

x = \pm 7

So $x = 7$ or $x = -7$ .

How to type your answer

Enter the two solutions separated by a comma. Order doesn’t matter. Examples: -2,-3, 7,-7, 5,-2. If both solutions are the same (a double root), enter the number twice: 3,3.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

x^2 = 9

Problem 2

x^2 = 25

Problem 3

x^2 = 100

Problem 4

(x - 4)(x + 2) = 0

Practice

Standard problems matching the lesson.

Problem 5

x^2 = 16

Problem 6

x^2 = 36

Problem 7

x^2 = 64

Problem 8

(x + 3)(x - 5) = 0

Problem 9

x^2 + 5x + 6 = 0

Problem 10

x^2 + 7x + 10 = 0

Problem 11

x^2 - 7x + 12 = 0

Problem 12

x^2 + 2x - 8 = 0

Problem 13

x^2 - x - 12 = 0

Problem 14

x^2 - 9 = 0

Problem 15

x^2 - 49 = 0

Problem 16

x^2 - 8x + 15 = 0

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

x^2 + 4x - 21 = 0

Problem 18

x^2 - 11x + 28 = 0

Problem 19

x^2 + 6x + 9 = 0

Problem 20

x^2 - 10x + 25 = 0

Problem 21

x^2 - 144 = 0

Problem 22

x^2 + 3x - 40 = 0