Basic Technical Mathematics, 11th Edition

7.3 The Quadratic Formula

• The Quadratic Formula • Character of the Roots of a Quadratic Equation

We now use the method of completing the square to derive a general formula that may be used for the solution of any quadratic equation.

Consider the general quadratic equation:

$$a{x}^2\hspace{0.17em}+\hspace{0.17em}bx\hspace{0.17em}+\hspace{0.17em}c\hspace{0.17em}=\hspace{0.17em}0(a\hspace{0.17em}\ne \hspace{0.17em}0)$$

When we divide through by a, we obtain

$$x^2\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{b}{a}}x\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{c}{a}}\hspace{0.17em}=\hspace{0.17em}0$$

Subtracting c/a from each side, we have

$$x^2\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{b}{a}}x\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}{\displaystyle \frac{c}{a}}$$

Half of b/a is b/2a, which squared is $b^2\hspace{0.17em}/\hspace{0.17em}4a^2\hspace{0.17em}.\hspace{0.17em}$ Adding $b^2\hspace{0.17em}/\hspace{0.17em}4a^2$ to each side gives us

$$x^2\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{b}{a}}x\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{b^2}{4a^2}}\hspace{0.17em}=\hspace{0.17em}\hspace{0.17em}-\hspace{0.17em}{\displaystyle \frac{c}{a}}\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{b^2}{4a^2}}$$

Writing the left side as a perfect square and combining fractions on the right side,

$${(x\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{b}{2a}})}^2\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{b^2\hspace{0.17em}-\hspace{0.17em}4ac}{4a^2}}$$

Equating $x\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{b}{2a}}$ to the positive and negative square root of the right side,

$$x\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{b}{2a}}\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{\hspace{0.17em}\pm \hspace{0.17em}\sqrt{b^2\hspace{0.17em}-\hspace{0.17em}4ac}}{2a}}$$

When ...