Factorising quadratics

Educational level: this is a secondary education resource.

Quadratic equations are equations of the form $ax^{2}+bx+c=0$ where a, b and c are constants, $a\neq 0$ and $x$ is a variable. In other words, a quadratic equation has at least one term of the variable, say $x$, raised to the exponent $2$, e.g. $x^{2}$

Subject classification: this is a mathematics resource.

Arranging terms

Arrange the quadratic into order: first the squared number ax², then the number times x, bx, finally the constant value c.

Factorising quadratics

Form of quadratics: $ax^{2}+bx+c=0$

To factorise:

1. split the middle term so it adds to the original number, e.g., let b = (AD + BC), and
2. multiplies to the constant times the first term, e.g., Ax times Bx equals ABx², then a = AB,
3. then bracket so the pronumeral (letter) is like this, e.g., (Ax + C)(Bx + D).

Checking

Multiplying the two terms: $(Ax+C)$ and $(Bx+D)$ with each other becomes:

$Ax\times Bx+Ax\times D+C\times Bx+C\times D$

which rearranges to:

$ABx^{2}+(AD+BC)x+CD$

The final constant $c=CD.$

Examples

$2m^{2}+11m+5$

$=(2m+1)(m+5)$

To check it, re-expand the answer to see if we get back to where we started from:

$(2M+1)(M+5)$

$=2M\times M+2M\times 5+1\times M+1\times 5$

$=2m^{2}+11m+5$

See also

• Complex Numbers

External links

{{Mathematics resources}}