## Quadratic Equations

### Introduction

A quadratic equation is defined as an equation of the form

y = a x ^{2} + b x + c

i.e. an equation consisting only of

• a y-term, but no y^{2} terms (or higher power y-terms)

• x-terms, the highest power of x being
represented by an x^{2} term.

• constant(s)

The above expression is the full general expression. An x^{2}-term is the only requirement for the equation to be classed as a quadratic - b and/or c can be zero and the equation would still be classed as a quadratic.

For example, all the following are still quadratics

y = x ^{2} + 3 x

y = x ^{2} + 2

y = 4 x ^{2}

### Completing the Square

Taking

y = x ^{2} + b x + c

we transform it to

\[ y = \left(x + \left(\frac{b}{2}\right)x\right)^2 - \left(\frac{b}{2}\right)^2 - c \]

If you compare the bracketed term with the first two terms of the original, you will see that the bracketed term is equal to the first two terms of the original, except for a $\left(\frac{b}{2}\right)^2$ term. If you therefore subtract this $\left(\frac{b}{2}\right)^2$ term (as we have done in the second equation), then the two equations will be identical. In a real-life example the $\left(\frac{b}{2}\right)^2$ term and c term can be simplified into just one term.

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Example__

Consider

y = x ^{2} + 6 x + 7

Converting this to

y = (x +3)^{2} -(3)^{2} + 7

and simplify it to

y = (x +3)^{2} -2

__
Example__

Consider

y = x ^{2} - 4 x + 12

Converting this to

y = (x -2)^{2} -(-2)^{2} + 7

and simplify it to

y = (x -2)^{2} + 3