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Toolkit 21

Quadratic Formula

ax2+bx+c=0  ⇒  x=−b±b2−4ac2aax^2+bx+c=0\;\Rightarrow\;x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Proof

Starting from

ax2+bx+c=0,ax^2+bx+c=0,

divide both sides by aa:

x2+bax+ca=0.x^2+\frac{b}{a}x+\frac{c}{a}=0.

Move the constant term to the right:

x2+bax=−ca.x^2+\frac{b}{a}x=-\frac{c}{a}.

Complete the square by adding b24a2\frac{b^2}{4a^2} to both sides:

(x+b2a)2=b24a2−ca=b2−4ac4a2.\left(x+\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2}-\frac{c}{a} = \frac{b^2-4ac}{4a^2}.

Taking square roots gives

x+b2a=±b2−4ac2a.x+\frac{b}{2a} = \pm\frac{\sqrt{b^2-4ac}}{2a}.

Therefore,

x=−b±b2−4ac2a.□x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}. \quad\square