Educational level: this is a secondary education resource.

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

 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.

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

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: ${\displaystyle (Ax+C)}$ and ${\displaystyle (Bx+D)}$ with each other becomes:

${\displaystyle Ax\times Bx+Ax\times D+C\times Bx+C\times D}$

which rearranges to:

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

The final constant ${\displaystyle c=CD.}$

Examples

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

${\displaystyle =(2m+1)(m+5)}$

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

${\displaystyle (2M+1)(M+5)}$

${\displaystyle =2M\times M+2M\times 5+1\times M+1\times 5}$

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