Difference between revisions of "Quadratic formula"

Revision as of 23:08, 31 October 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 complex solutions to the quadratic equation.

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

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

@@ Line 25: / Line 25: @@
 Given the values <math>{a},{b},{c}</math>, we can find all [[real]] and [[complex number|complex]] solutions to the quadratic equation.
+=== Variation ===
+<math>\frac{2c}{-b\pm\sqrt{b^2-4ac}}</math>
+In some situations, it is preferable to use this variation of the quadratic formula.