Solving Quadratic Equations by Factoring

Quadratic equations are equations that contain a squared variable term, such as x². One common method for solving quadratic equations is factoring.

Factoring involves breaking down the quadratic expression into simpler factors that can be set equal to zero.

In this article, we will discuss solving quadratic equations by factoring.

Steps for Factoring to Solve Quadratics

Solving quadratic equations by factoring involves four main steps:

• Put the equation in standard quadratic form: ax² + bx + c = 0
• Factor the quadratic expression ax² + bx + c
• Set each factor equal to zero and solve the simpler equations
• Verify the solutions by plugging them back into the original equation

Example

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

Step 1) Put Into Standard Form

The equation is already in standard quadratic form, with a = 1, b = -5, and c = 6.

Step 2) Factor the Quadratic Expression

x² – 5x + 6 = x² – 3x -2x + 6

=(x – 2)(x – 3)

Step 3) Set Factors Equal to Zero

Setting each factor equal to zero gives us:

(x – 2) = 0 , (x – 3) = 0

x – 2 = 0 , x – 3 = 0

x = 2 , x = 3

The possible solutions are x = 2 and x = 3.

Step 4) Verify Solutions

Plugging x = 2 and x = 3 back into the original equation.

Therefore, the solutions are x = 2 and x = 3.

Solved Examples Quadratic Equations by Factoring

Example

Solve 3x² – 5x – 2 = 0

Solution

3x² – 5x – 2 = 0

3x² – 6x + x – 2 = 0

3x (x – 2) + 1 (x – 2) = 0

(3x + 1)(x – 2) = 0

3x + 1 = 0 , x – 2 = 0

x = -1/3 , x = 2

Solutions: x = -1/3 and x = 2

Example

Solve x² + x – 20 = 0

Solution

x² + x – 20 = 0

x² + 5x -4x – 20 = 0

x (x + 5) -4 (x + 5) = 0

(x + 5)(x – 4) = 0

(x + 5) = 0 , (x – 4) = 0

x = -5 , x = 4

Solutions: x = 4 and x = -5

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