# Expand And Factorize Quadratic Expressions

## Expanding Quadratic Expressions:

**Quadratic expressions** are algebraic expressions where the **variable has a power of 2**.

For example: x^{2} + 3x + 4

To expand quadratic equations, use the **FOIL (First, Outside, Inside, Last)** method.

First
Outside
Inside
Last

## Example One

Expand ( x + 3 ) ( x + 2 ) without and with using FOIL.

**Answer (without using FOIL):**

( x + 3 ) ( x + 2 )

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

= x^{2} + 2x + 3x + 6

= x^{2} + 5x + 6

**Answer (with using FOIL):**

( x + 3 ) ( x + 2 )

= x^{2} + 2x + 3x + 6

= x^{2} + 5x + 6

## Example Two

( x + 4 ) ( x – 2 )

= x^{2} – 2x + 4x – 8

= x^{2} + 2x – 8

## Example Three

( 2x + 5 ) ( 3x – 8 )

= 6x^{2} – 16x + 15x – 40

= 2x^{2} – x – 40

## Questions - Expand Using FOIL

**Q1.** ( x + 6 ) ( x + 5 )

**Q2.** ( x – 5 ) ( x – 4 )

**Q3.** ( 2x + 5 ) ( 6x – 2 )

**Answers**

**A1.** x^{2} + 11x + 30

**A2.** x^{2} – 9x + 20

**A3.** 12x^{2} + 26x – 10

## Perfect Squares

( x + a )^{2} = x^{2} + 2ax + a^{2}

( x – a )^{2} = x^{2} – 2ax + a^{2}

## Example Four

( x + 5 )^{2}

= ( x + 5 ) ( x + 5 )

= x^{2} + 10x + 25

## Example Five

( x – 3 )^{2}

= ( x – 3 ) ( x – 3 )

= x^{2} – 6x + 9

## Questions - Expand These Perfect Squares

**Q1.** ( x + 7 )^{2}

**Q2.** ( 2x + 5 )^{2}

**Answers**

**A1.** x^{2} + 14x + 49

**A2.** 4x^{2} + 20x + 25

## Difference of Squares

( x + a ) ( x – a ) = x^{2} – a^{2}

## Example Six

( x + 5 ) ( x – 5 )

= x^{2} – 5x + 5x – 25

= x^{2} – 25

## Example Seven

( x – 3 ) ( x + 3 )

= x^{2} – 3x + 3x – 9

= x^{2} – 9

## Questions - Expand These Difference of Squares

**Q1.** ( x + 7 ) ( x – 7 )

**Q2.** ( 2x + 5 ) ( 2x – 5 )

**Answers**

**A1.** x^{2} – 49

**A2.** 4x^{2} – 25

## Factorizing Quadratic Expressions:

Factorizing is the **reverse of expanding**.

## Example Eight

x^{2} + 6x + 5

= ( x + 5 ) ( x + 1 )

## Example Nine

6x^{2} + 2x – 20

= ( 2x + 4 ) ( 3x – 5 )

## Questions - Factorize

**Q1.** x^{2} – 7x – 8

**Q2.** x^{2} + x – 12

**Answers**

**A1.** ( x – 8 ) ( x + 1 )

**A2.** ( x + 4 ) ( x – 3 )