Solving Quadratic Equations by Quadratic Formula

Quadratic Formula

Quadratic Formula

In many cases, factorising a quadratic equation or completing the square cane be long or difficult. We can instead use the quadratic formula.

\( \begin{align} \displaystyle \require{color} ax^2 + bx + c &= 0 \\ ax^2 + bx &= -c \\ x^2 + \dfrac{b}{a}x &= -\dfrac{c}{a} \\ x^2 + \dfrac{b}{a}x \color{red} + \Big(\dfrac{b}{2a}\Big)^2 &= -\dfrac{c}{a} \color{red} + \Big(\dfrac{b}{2a}\Big)^2\\ \Big(x+\dfrac{b}{2a}\Big)^2 &= \dfrac{b^2-4ac}{4a^2} \\ x+\dfrac{b}{2a} &= \pm \sqrt{\dfrac{b^2-4ac}{4a^2}} \\ x &= -\dfrac{b}{2a} \pm \dfrac{\sqrt{b^2-4ac}}{2a} \\ \therefore x &= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} \\ \end{align} \)

Example 1

Solve $x^2 - 2x - 6 =0$ for $x$.

Example 2

Solve $2x^2 + 3x - 6 = 0$ for $x$.