QUADRATIC EQUATIONS

LEARNING OBJECTIVE

Students will be able to solving quadratic equations algebraically use variety methods (Factorising, Completing Square, and General Formula)

INTRODUCTION

The study of

**algebra**is vital for many areas of mathematics. We need it to manipulate equations, solve problems from unknown variables, and also to develop higher level mathematical theories.

A

**QUADRATIC EQUATION**is an equation which can be written in the form ax2 + bx + c = 0 where a, b, and c are constants and a is not equal to zero.

A quadratic equation may have

*two, one or zero real solutions.*

SOLUTION BY FACTORISATION

For quadratic equations which are not of the form x2 = k, we need an alternative method of solution. One method is to FACTORISE the quadratic and then apply the NULL FACTOR LAW.

The Null Factor Law states that :

*When the product of two or more numbers is zero, then at least one of them must be zero.*

*so if ab = 0 then a = 0 or b = 0*

To solve quadratic equations by factorisation, we follow these steps:

Step 1 : If necessary, rearrange the equation so one side is

**ZERO**

Step 2 :

**Fully Factorise**the other side (usually the LHS/Left Hand Side)

Step 3 : Apply the

**Null Factor Law**

Step 4 :

**Solve**the resulting linear equations

Example

Solution by Factorisation |

COMPLETING THE SQUARE

Some quadratic equations, such as x^2+4x-7=0, cannot be solved using the factorisation methods already practised. This is because the solutions are irrational.

Instead, we use a method called c

**ompleting the square.**

Completing the Square |

**perfect square**on the left hand side is called

**completing the square**. To complete a perfect square, the number we must add to both sides is found by

**halving the coefficient of x, then squaring this value.**

**THE QUADRATIC FORMULA**

Many quadratic equations cannot be solved easily by factorisation, and completing the square is rather tedious. Consequently, the quadratic formula has been developed.

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