How Do You Solve a Quadratic Equation by Factoring; Completing the Square; or the Quadratic Formula?

A quadratic equation can be solved by factoring, completing the square, or applying the quadratic formula. First rearrange it into standard form, $ax^2+bx+c=0$, then select the most efficient valid method.

This is IB Mathematics: Analysis and Approaches, SL 2.7, and the analytic methods are particularly relevant to Paper 1.

Before choosing a method, identify $a$, $b$, and $c$ carefully, including their signs. Factoring searches for two expressions whose product reproduces the quadratic. Completing the square rewrites the equation using a perfect square, while the formula gives the roots directly from the coefficients.

Consider $x^2-5x+6=0$:

For completing the square, divide by $a$ first if $a\neq1$, then add the square of half the coefficient of $x$ to both sides.

The discriminant is $\Delta=b^2-4ac$. If $\Delta>0$, there are two distinct real roots; if $\Delta=0$, there is one repeated real root; if , there are no real roots.

Always substitute each proposed root into the original equation when checking. For example, $2^2-5(2)+6=0$ and $3^2-5(3)+6=0$, so both values are valid. This check can reveal sign errors, but it does not replace the algebraic working required by the command term in the examination.

A common misconception is dropping the $\pm$ when taking a square root. This loses one solution and therefore loses an accuracy mark.

Exam technique: For the command term “find”, show relevant stages. On Paper 1, keep answers exact; on Paper 2, a GDC may check solutions, but you should still show the equation and method used.