Difference between revisions of "Quadratic formula"

Revision as of 18:25, 12 December 2021

The quadratic formula is a general expression for the solutions to a quadratic equation. It is used when other methods, such as completing the square, factoring, and square root property do not work or are too tedious.

Let the quadratic be in the form $ax^2+bx+c=0$ .

Moving $c$ to the other side, we obtain

$ax^2+bx=-c.$

Dividing by ${a}$ and adding $\frac{b^2}{4a^2}$ to both sides yields

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

Completing the square on the left-hand side gives

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

As described above, an equation in this form can be solved, yielding

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

This formula is also called the quadratic formula.

Given the values ${a},{b},{c},$ we can find all real and complex solutions to the quadratic equation.

In some situations, it is preferable to use this variation of the quadratic formula:

$\frac{2c}{-b\pm\sqrt{b^2-4ac}}$

@@ Line 6: / Line 6: @@
 Moving <math>c</math> to the other side, we obtain
-<math>ax^2+bx=-c</math>
+<cmath>ax^2+bx=-c.</cmath>
 Dividing by <math>{a}</math> and adding <math>\frac{b^2}{4a^2}</math> to both sides yields
-<math>x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}</math>.
+<cmath>x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}.</cmath>
 Completing the square on the left-hand side gives
-<math>\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}</math>
+<cmath>\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}.</cmath>
 As described above, an equation in this form can be solved, yielding
-<math>{x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}}</math>
+<cmath>{x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}}.</cmath>
 This formula is also called the quadratic formula.