Solving Quadratic Equations by Factoring

Solving Quadratic Equations by

Factoring

Quadratic Equations Quadratic equations all follow a specific form.

ax  bx  c  0 2

Quadratic Equations Quadratic equations all follow a specific form.

2 x  9 x  36  0 2

3x – 4x  8  0

An x2 term must be included

Quadratic Equations Quadratic equations all follow a specific form.

2 x  9 x  36  0 2

3x 2 – 4 x 3  8  0

No powers of x higher than 2

Quadratic Equations Quadratic equations all follow a specific form.

2 x  9 x  36  0 2

3x 2 – 4 x  8  3x 2

When simplified, the equation is not linear

Solutions to Quadratic Equations To solve a quadratic equation, we must find all the values for x for which equality holds; these values are called the roots of the equation.

x  3 x  10  0 2

52  3(5)  10  0 2 (2)  3(2)  10  0

5 and –2 are roots of this equation.

The solution to every quadratic equation always consists of two roots.

Solutions to Quadratic Equations There are three possible types of solutions when solving quadratic equations.

Imaginary Roots

Double Roots

Real Roots

x – 6x  8  0 x – 6x  9  0 2

x 1  0

2

Solution: x = 2, x = 4

Solution: x = 3, x = 3

2

Solution: x=

– 𝟏, x = – – 𝟏

Solving Quadratic Equations by Factoring Step 1: Simplify the quadratic equation (if you can).

x  2 x – 24  0 2

Solving Quadratic Equations by Factoring Step 2: Factor the quadratic expression.

x  2 x – 24  0 ( x – 4)( x  6)  0 2

Solving Quadratic Equations by Factoring Step 3: Set each factor equal to zero.

x  2 x – 24  0 ( x – 4)( x  6)  0 2

( x – 4)  0

( x  6)  0

Solving Quadratic Equations by Factoring Step 4: Solve each linear equation. The two solutions are the roots of the quadratic equations.

x  2 x – 24  0 ( x – 4)( x  6)  0 ( x  6)  0 ( x – 4)  0 x  –6 x4 2

Check solutions in the original equation.

Solving Quadratic Equations by Factoring Check solutions in the original equation

x  –6

x4

x  2 x – 24  0

x  2 x – 24  0

2

2

(4)  2(4) – 24  0 (–6)  2(–6) – 24  0 2

2

16  8 – 24  0

36  (–12) – 24  0

00

00

Solve the following equation:

𝟏𝟔𝒙𝟐 – 𝟖𝒙 + 𝟏 = 𝟎

Solve the following equation:

𝟏𝟔𝒙𝟐 – 𝟖𝒙 + 𝟏 = 𝟎 Checking Solutions

Simplify following expression: Solve thethe following equation:

3x2 + 30 = 21x a. b. c. d.

x = 2, x = 5 x = –2, x = –5 x = 5, x = 6 x = –5, x = –6

Solve the following equation:

3x2 + 30 = 21x

Solve the following equation:

3x2 + 30 = 21x Checking Solutions