How to Factor Quadratics (Step by Step)

Before anything else, look for a greatest common factor (GCF) that every term shares. Pulling it out makes the rest of the problem smaller and easier.

Example: Factor $6x^2 + 9x$.

1. Both coefficients are divisible by 3, and both terms contain $x$. So the GCF is $3x$.
2. Divide each term by $3x$: $6x^2 \div 3x = 2x$ and $9x \div 3x = 3$.
3. Write the result: $6x^2 + 9x = 3x(2x + 3)$.

The GCF can also be just a number. For $2x^2 + 14x + 24$, pull out 2 to get $2(x^2 + 7x + 12)$, then factor the trinomial inside.

For $x^2 + bx + c$, you need two numbers that multiply to $c$ and add to $b$.

Example: Factor $x^2 + 7x + 12$.

1. List the factor pairs of 12: $1 \cdot 12$, $2 \cdot 6$, $3 \cdot 4$.
2. Find the pair that adds to 7: $3 + 4 = 7$.
3. Write the factors: $(x + 3)(x + 4)$.
4. Check by expanding: $x^2 + 4x + 3x + 12 = x^2 + 7x + 12$. It matches.

Example with negatives: Factor $x^2 - 2x - 15$.

1. You need a product of $-15$ and a sum of $-2$.
2. A negative product means one number is positive and one is negative. Try $-5$ and $3$: $(-5)(3) = -15$ and $-5 + 3 = -2$.
3. Answer: $(x - 5)(x + 3)$.

Sign shortcut: if $c$ is positive, both numbers have the same sign as $b$. If $c$ is negative, the numbers have opposite signs.

When the leading coefficient is not 1, as in $ax^2 + bx + c$, use the AC method (also called factoring by grouping).

Example: Factor $2x^2 + 7x + 3$.

1. Multiply $a \cdot c$: $2 \cdot 3 = 6$.
2. Find two numbers that multiply to 6 and add to $b = 7$. That is 1 and 6.
3. Split the middle term using those numbers: $2x^2 + x + 6x + 3$.
4. Group in pairs and factor each pair: $x(2x + 1) + 3(2x + 1)$.
5. Factor out the shared binomial: $(2x + 1)(x + 3)$.

Example: Factor $3x^2 - 10x - 8$.

1. $a \cdot c = 3 \cdot (-8) = -24$.
2. Two numbers with a product of $-24$ and a sum of $-10$: $-12$ and $2$.
3. Split: $3x^2 - 12x + 2x - 8$.
4. Group: $3x(x - 4) + 2(x - 4)$.
5. Answer: $(3x + 2)(x - 4)$.

If the two groups in step 4 do not share the same binomial, recheck your signs.

$$a^2 - b^2 = (a - b)(a + b)$$

Look for two perfect squares separated by a minus sign, with no middle term.

• $x^2 - 49 = (x - 7)(x + 7)$
• $4x^2 - 25 = (2x - 5)(2x + 5)$, because $4x^2 = (2x)^2$ and $25 = 5^2$.

A sum of squares, like $x^2 + 9$, does not factor using real numbers.

$$a^2 + 2ab + b^2 = (a + b)^2 \qquad a^2 - 2ab + b^2 = (a - b)^2$$

The first and last terms are perfect squares, and the middle term is twice the product of their square roots.

Example: Factor $9x^2 - 12x + 4$.

1. $9x^2 = (3x)^2$ and $4 = 2^2$.
2. Twice the product: $2 \cdot 3x \cdot 2 = 12x$. That matches the middle term.
3. The middle sign is negative, so the answer is $(3x - 2)^2$.

A quicker one: $x^2 + 10x + 25 = (x + 5)^2$, since $2 \cdot x \cdot 5 = 10x$.

1. Pull out the GCF.
2. Two terms with a minus sign? Try difference of squares.
3. Three terms? Check for a perfect square trinomial first.
4. If $a = 1$, find two numbers that multiply to $c$ and add to $b$.
5. If $a \neq 1$, use the AC method.
6. Expand your answer to check it.

Paste it into the VeilGPT solver to check your steps if you get stuck.