Factorising calculator for quadratics

Find two numbers that multiply to $c$ and add to $b$.

Example: $x^2 + 5x + 6$. $2 \times 3 = 6$ and $2 + 3 = 5$, so $(x + 2)(x + 3)$.

Check by expanding: $x^2 + 3x + 2x + 6 = x^2 + 5x + 6$. Our expanding brackets calculator does that check.

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

1. Multiply $a \times c = 6$. Find two numbers that multiply to 6 and add to 7: 1 and 6.
2. Split the middle term: $2x^2 + x + 6x + 3$.
3. Group: $x(2x + 1) + 3(2x + 1)$.
4. Factorise: $(2x + 1)(x + 3)$.

Another: $6x^2 - x - 2 = (3x - 2)(2x + 1)$.

• Difference of two squares: $x^2 - 9 = (x - 3)(x + 3)$. In general, $a^2 - b^2 = (a - b)(a + b)$.
• No constant term: $3x^2 + 6x = 3x(x + 2)$: take out the common factor.
• Common factor first: $2x^2 + 4x - 6 = 2(x^2 + 2x - 3) = 2(x - 1)(x + 3)$.

If $(x + 2)(x + 3) = 0$, one bracket must be 0, so $x = -2$ or $x = -3$. Notice the signs flip. When a quadratic doesn't factorise, use the quadratic equation solver (formula) or the completing the square calculator.

• Finding numbers that multiply to $b$ instead of $c$.
• Sign slips: if $c$ is negative, one number is positive and one negative.
• Not taking out a common factor first.

Practise with our factorising questions.

• Always expand your answer to check it.
• "Factorise fully" means take out every common factor, including numbers.
• If the question then asks you to solve, set each bracket equal to 0.

Factorising means writing an expression as a product. Before quadratics, you'll meet:

• Single brackets: $6x + 9 = 3(2x + 3)$, taking out the highest common factor.
• Letters as well as numbers: $4x^2y - 6xy^2 = 2xy(2x - 3y)$.
• Four terms: $ax + ay + bx + by = (a + b)(x + y)$, by grouping.

The same idea runs through all of them: find what every term has in common and take it outside the bracket.