Introduction

The quadratic formula is a formula in getting the roots of the quadratic equation $ax^2 + bx + c = 0$, where $a$, $b$ and $c$ are real numbers and $a \neq 0$. The quadratic formula is a generalization of completing the square, and it is usually used as a calculation strategy if a quadratic equation is not factorable.

In the graph of the quadratic function $f(x) = ax^2 + bx + c$, the values of $x$ in $ax^2 + bx + c = 0$, are the coordinates of $x$ where the curve pass through.

But how did mathematicians come up with the quadratic formula? How were they able to invent such complicated formula? In this post, we discuss the derivation of the quadratic formula.

Theorem

If $ax^2 + bx + c = 0$ where $a \neq 0$, then

$x = \displaystyle \frac{-b \pm \sqrt{b^2 - 4ac} }{2a}$

Proof

Given: $ax^2 + bx + c = 0$ and $a \neq 0$.

Divide both sides by $a$.

$\displaystyle x^2 + \frac{b}{a}x + \frac{c}{a} = 0$.

Add $\displaystyle \frac{-c}{a}$ to each side.

$\displaystyle x^2 + \frac{b}{a}x = \frac{-c}{a}$.

Complete the square by getting half of $\frac{b}{a}$, squaring it and then adding the result to both sides.

$x^2 + \displaystyle \frac{b}{a}x + \frac{b^2}{4a^2} = \frac{b^2}{4a^2} - \frac{c}{a}$.

Write the left side as a binomial squared.

$\left ( x + \displaystyle \frac{b}{2a} \right )^2 = \displaystyle \frac{b^2}{4a^2} - \frac{c}{a}$

Simplify the right hand side of the equation.

$\left ( x + \displaystyle \frac{b}{2a} \right )^2 = \displaystyle \frac{b^2-4ac}{4a^2}$

Take the square root of both sides.

$x + \displaystyle \frac{b}{2a} = \frac{\pm \sqrt{b^2 - 4ac}}{2a}$.

Add $\frac{-b}{2a}$ to both sides.

$x = \displaystyle\frac{ -b \pm \sqrt{b^2 - 4ac}}{2a} \blacksquare$.

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