1. When a = 1 (Easy Case)

Example:
$x^2+7x+10$x2+7x+10

Use this idea: Find two numbers that multiply to $c$c and add to $b$b.

Here:

• Multiply to 10
• Add to 7

Numbers: 5 and 2

So the factorization is:

$(x+5)(x+2)$(x+5)(x+2)

Check:
5 2 = 10
5 + 2 = 7

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2. When a 1 (AC Method)

Example:
$2x^2+7x+3$2x2+7x+3

Step 1: Multiply a c

2 3 = 6

Step 2: Find two numbers

• Multiply to 6
• Add to 7

Numbers: 6 and 1

Step 3: Rewrite the middle term

$2x^2+6x+x+3$2x2+6x+x+3

Step 4: Factor by grouping

$(2x^2+6x)+(x+3)$(2x2+6x)+(x+3)

Factor each group:

$2x(x+3)+1(x+3)$2x(x+3)+1(x+3)

Step 5: Final factor

$(2x+1)(x+3)$(2x+1)(x+3)

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3. Quick Pattern Method (for $x^2+bx+c$x2+bx+c)

If the first term is $x^2$x2, you can use this shortcut:

$x^2+bx+c=(x+\_)(x+\_)$x2+bx+c=(x+_)(x+_)

Just find numbers that:

• Multiply = c
• Add = b

Example:

$x^2+9x+20$x2+9x+20

Numbers that multiply to 20 and add to 9:

4 and 5

$(x+4)(x+5)$(x+4)(x+5)

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4. Always Check Your Answer

Multiply the factors again (FOIL method).

Example:

$(x+4)(x+5)$(x+4)(x+5)

FOIL:

• First: $x^2$x2
• Outer: $5x$5x
• Inner: $4x$4x
• Last: $20$20

$x^2+9x+20$x2+9x+20

Correct

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5. Tips to Factor Faster

Look for common factors first
Example:

$2x^2+10x+12$2x2+10x+12

Factor 2 first:

$2(x^2+5x+6)$2(x2+5x+6)

Then factor inside:

$2(x+2)(x+3)$2(x+2)(x+3)

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Memorize multiplication pairs

Example:

12 (1,12) (2,6) (3,4)
20 (1,20) (2,10) (4,5)

This makes factoring much faster.

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Example Practice

1. $x^2+8x+15$x2+8x+15

Numbers: 3 and 5

$(x+3)(x+5)$(x+3)(x+5)

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1. $x^2+11x+24$x2+11x+24

Numbers: 3 and 8

$(x+3)(x+8)$(x+3)(x+8)

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1. $3x^2+10x+3$3x2+10x+3

Multiply a c

3 3 = 9

Numbers: 9 and 1

Final:

$(3x+1)(x+3)$(3x+1)(x+3)

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