Difference between revisions of "Quadratic formula"

Revision as of 11:19, 19 June 2006

The quadratic formula is a general expression for the solutions to a quadratic equation.

Let the quadratic be in the form $a\cdot x^2+b\cdot x+c=0$ .

Moving c to the other side, we obtain

$a\cdot x^2+b\cdot x=-c$

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

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

Factoring the LHS 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 nonreal solutions to the quadratic equation.

@@ Line 10: / Line 10: @@
 <math>a\cdot x^2+b\cdot x=-c</math>
-Dividing by <math>a</math> and adding <math>\frac{b^2}{4a^2}</math> to both sides yields
+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>.