Before you read

Answer first — see what you already know.

A laboratory models a chemical reaction rate as $2 x^{2} + 5 x + 3$ . What is the factored form?

A
$(2 x + 3) (x + 1)$
B
$(2 x + 1) (x + 3)$
C
$(2 x - 1) (x - 3)$
D
Cannot be factored

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A structural formula involves $3 x^{2} - 7 x + 2$ . Which is the correct factorization?

A
$(3 x - 1) (x - 2)$
B
$(3 x + 1) (x - 2)$
C
$(3 x - 2) (x - 1)$
D
$(x - 1) (3 x + 2)$

Check your answer

Answer:

Theory

The AC Method (for ax $^{2}$ + bx + c)

When $a \neq 1$ , factoring is harder. Use the AC method:

1. Multiply $a \times c$
2. Find two numbers that multiply to $ac$ and add to $b$
3. Rewrite the middle term using those two numbers
4. Factor by grouping

$2x^2 + 7x + 3$
$2x^2 + 6x + x + 3$ split middle term
$2x(x+3) + 1(x+3)$ factor by grouping
$(2x+1)(x+3)$

Factoring by grouping

Example 1

Factor $2x^2 + 7x + 3$

AC: $a \times c = 2 \times 3 = 6$

Find: two numbers that multiply to $6$ and add to $7$ : that's $1$ and $6$ .

Rewrite: $2x^2 + 1x + 6x + 3$

Group: $(2x^2 + x) + (6x + 3)$

Factor each group: $x(2x + 1) + 3(2x + 1)$

Factor out $(2x + 1)$ : $(2x + 1)(x + 3)$

(2x + 1)(x + 3)

(2x+1)(x+3) = 2x^2 + 6x + x + 3 = 2x^2 + 7x + 3

✓

Example 2

Factor $3x^2 - 10x + 8$

AC: $3 \times 8 = 24$

Find: multiply to $24$ , add to $-10$ : $-4$ and $-6$ .

Rewrite: $3x^2 - 4x - 6x + 8$

Group: $(3x^2 - 4x) + (-6x + 8)$

Factor: $x(3x - 4) - 2(3x - 4)$

Result: $(3x - 4)(x - 2)$

(3x - 4)(x - 2)

Theory

The Trial-and-Error Method

For small coefficients, guessing and checking can be faster:

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

The first blanks must multiply to $a$ , the last blanks to $c$ . Try combinations until the cross-products add to $b$ .

Example: $6x^2 + 11x + 4$
- $6 = 6 \times 1$ or $3 \times 2$
- $4 = 4 \times 1$ or $2 \times 2$
- Try $(3x + 4)(2x + 1)$ : $3x \cdot 1 + 4 \cdot 2x = 3x + 8x = 11x$ ✓

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The AC Method (for ax2^{2}2 + bx + c)

The Trial-and-Error Method

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The AC Method (for ax $^{2}$ + bx + c)