Factorising Quadratics (DP IB Analysis & Approaches (AA): HL): Revision Note

Factorising quadratics

Why is factorising quadratics useful?

• Factorising gives roots (zeroes or solutions) of a quadratic

• It gives the $x$-intercepts of the graph of the quadratic

How do I factorise a monic quadratic of the form x²+bx+c?

• A monic quadratic is a quadratic where the coefficient of the $x^2$ term is 1

• You might be able to spot the factors by inspection

  • Especially if $c$ is a prime number

• Otherwise, start by finding two numbers $m$ and $n$ which have,

  • A sum equal to $b$

    • $p+q=b$

  • A product equal to $c$

    • $pq=c$

• Rewrite the middle term of the quadratic, $bx$, as $mx+nx$

• Use this to factorise $x^2+mx+nx+c$

• A shortcut is to write down $(x+p)(x+q)$ as soon as you have found $p$ and $q$

How do I factorise a non-monic quadratic of the form ax²+bx+c?

• A non-monic quadratic is a quadratic where the coefficient of the $x^2$ term is not equal to 1

• If $a$, $b$ and $c$ have a common factor then first factorise that out to leave a quadratic with coefficients that have no common factors

  • E.g. $3x^2+6x-45$ can be rewritten as $3\left(x^2+2x-15\right)$

• You might be able to spot the factors by inspection

  • Especially if $a$ and/or $c$ are prime numbers

• Otherwise, start by finding two numbers $m$ and $n$ which have,

  • A sum equal to $b$

    • $m+n=b$

  • A product equal to $ac$

    • $mn=ac$

• Rewrite the middle term, $bx$, as $mx+nx$

• Use this to factorise $a{x}^2+mx+nx+c$

• A shortcut is to write

  • $\frac{(ax+m)(ax+n)}{a}$

  • Then factorise common factors from numerator to cancel with the a on the denominator

How do I use the difference of two squares to factorise a quadratic of the form a²x²-c²?

• The difference of two squares can be used when

  • There is no linear $x$ term and

  • The constant term is a negative

  • E.g. $9x^2-16$

• First, square-root the two terms $a^2{x}^2$ and $c^2$

• The two factors of the quadratic are the sum of the square roots and the difference of the square roots

  • i.e. $\left(ax+c\right)\left(ax-c\right)$

  • $9x^2-16=\left(3x+4\right)\left(3x-4\right)$