CraigsMaths

Quadratic general to standard form

CraigsMaths
Mathematics Teaching for Learning
Engineering Maths B, Quadratic Equations
August 6th, 2008 by Craig Rose
Permalink: http://www.craigsmaths.com/ea003-engineering-mathematics-b/quadratic-general-to-standard-form/

When solving quadratic equations and sketching parabolas you can get the required points onto your graph very quickly if you know how to convert from the general form of $y=ax^{2}+bx+c$ to the standard form of $y=a\left(x-h\right)^{2}+k$ . (See Standard Quadratic Form for the reasons why.) This article explains how to complete the square to acheive this.

Example

We will convert the quadratic equation

$y=2x^{2}-32x+39$

into the standard form

$y=a\left(x-h\right)^{2}+k$

by completing the square.

Step 1

Rearrange your quadratic so that it is in the general form $y=ax^{2}+bx+c$

$y=2x^{2}-32x+39$

Step 2

Divide the coefficients of the $x^{2}$ and the $x$ terms by a number that will leave the $x^{2}$ term with a coefficient of $1$ .

In this case we divide by 2.

$y=2\left(x^{2}-16x\right)+39$

Step 3

Add the square of half the coefficient of the $x$ term to the grouped $x$ and $x^{2}$ terms. Subtract this amount times the constant in front of the brackets from the constant term.

In this case we want to add:

$\left(\frac{16}{2}\right)^{2}=64$