## You are here

Homederivation of quadratic formula

## Primary tabs

# derivation of quadratic formula

Suppose $A,B,C$ are real numbers, with $A\neq 0$, and suppose

$Ax^{2}+Bx+C=0.$ |

Since $A$ is nonzero, we can divide by $A$ and obtain the equation

$x^{2}+bx+c=0,$ |

where $b=\frac{B}{A}$ and $c=\frac{C}{A}$. This equation can be written as

$x^{2}+bx+\frac{b^{2}}{4}-\frac{b^{2}}{4}+c=0,$ |

so completing the square, i.e., applying the identity $(p+q)^{2}=p^{2}+2pq+q^{2}$, yields

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

Then, taking the square root of both sides, and solving for $x$, we obtain the solution formula

$\displaystyle x$ | $\displaystyle=$ | $\displaystyle-\frac{b}{2}\pm\sqrt{\frac{b^{2}}{4}-c}$ | ||

$\displaystyle=$ | $\displaystyle\frac{B}{2A}\pm\sqrt{\frac{B^{2}}{4A^{2}}-\frac{C}{A}}$ | |||

$\displaystyle=$ | $\displaystyle\frac{-B\pm\sqrt{B^{2}-4AC}}{2A},$ |

and the derivation is completed.

A slightly less intuitive but more aesthetically pleasing approach to this derivation can be achieved by multiplying both sides of the equation

$\displaystyle ax^{2}+bx+c=0$ |

by $4a$, resulting in the equation

$\displaystyle 4a^{2}x^{2}+4abx+b^{2}=b^{2}-4ac,$ |

in which the left-hand side can be expressed as $(2ax+b)^{2}$. From here, the proof is identical.

Related:

QuadraticFormula, QuadraticEquationInMathbbC

Type of Math Object:

Proof

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

12D10*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Oct 21

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag