Classwork Series And Exercises {JSS3- Mathematics}: Factorisation Of Quadratic Expressions

JSS 3 Mathematics Second Term Week 1

Topic: FACTORISATION OF QUADRATIC EXPRESSIONS

You can also factorise quadratic expressions. Remember that factorizing an expression simplifies it in some way. Factorizing is the reverse of expanding brackets.

Factorizing Quadratic Expressions

Multiply these brackets to remind yourself how to factorise.

( x + 2 ) ( x + 5 ) = x² + 7x + 10

( x + 2 ) ( x + 3 ) = x² + 5x + 6

( x – 3 ) ( x – 5 ) = x ² – 8x + 15

( x + 6 ) ( x – 5 ) = x² + x – 30

( x – 6 ) ( x + 5 ) = x² – x – 30

Factorising

To factorise an expression such x² + 5x + 6, you need to look for two numbers that add up to make 5 and multiply to give 6.

The factor pairs of 6 are:

1 and 6

2 and 3

2 and 3 add up to 5. So: (x +2) (x+3) = x² + 5x + 6

Factorising expressions gets trickier with negative numbers.

Example

Factorise the expression: c²– 3c – 10

Write down the expression: c²– 3c – 10

Remember that to factorise an expression we need to look for common factor pairs. In this example we are looking for two numbers that:

multiply to give -10

add to give -3

Think of all the factor pairs of -10:

1 and -10

-1 and 10

2 and -5

-2 and 5

Which of these factor pairs can be added to get -3?

Only 2 + (-5) = -3

So the answer is:

c² – 3c – 10 = (c + 2)(c – 5)

Factorising the difference of two squares

Some quadratic expressions have only a term in x² and a number such as x² – 25.

These quadratic expressions have no x term.

Using our method to factorise quadratics means we look for two numbers that multiply to make -25 and add to make 0.

The only factor pair that will work are 5 and -5. So:

(x + 5)(x – 5) = x² – 25

Not all quadratic expressions without an x term can be factorised.

Examples

Factorise:

x² – 4 = (x + 2)(x – 2)

x² – 81 = (x + 9)(x – 9)

x² – 9 = (x + 3)(x – 3)

Solving Quadratic Equations by Factorising

To solve a quadratic equation, the first step is to write it in the form: ax² + bx + c = 0. Then factorise the equation as you have revised in the previous section.

If we have two numbers, A and B, and we know that A × B = 0, then it must follow that either A = 0, or B = 0 (or both). When we multiply any number by 0, we get 0.

Example

Solve the equation: x² – 9x + 20 = 0

Solution

First, factorise the quadratic equation x²– 9x + 20 = 0

Find two numbers which add up to 9 and multiply to give 20. These numbers are 4 and 5.

(x – 4) (x – 5) = 0

Now find the value x so that when these brackets are multiplied together the answer is 0.

This means either (x – 4) = 0 or (x – 5) = 0

So x = 4 or x = 5.

You can check these answers by substituting 4 and 5 in to the equation:

x²– 9x + 20

Substituting 4 gives:

4² – 9 × 4 + 20 = 16 – 36 + 20 = 0

Substituting 5 gives:

5² – 9 × 5 + 20 = 25 – 45 + 20 = 0

Remember these 3 simple steps and you will be able to solve quadratic equations.

Now try this question.

Completing the square

This is another way to solve a quadratic equation if the equation will not factorise.

It is often convenient to write an algebraic expression as a square plus another term. The other term is found by dividing the coefficient of x by 2, and squaring it.

Any quadratic equation can be rearranged so that it can be solved in this way.

Have a look at this example.

Example

Rewrite x² + 6x as a square plus another term.

The coeffient of x is 6. Dividing 6 by 2 and squaring it gives 9.

x² + 6x = (x² + 6x + 9) – 9

= (x + 3)² – 9

Example

We have seen in the previous example that x² + 6x = (x + 3)² – 9

So work out x² + 6x – 2

x² + 6x – 2 = ( x² + 6x + 9 ) – 9 – 2 = (x + 3)² – 11

Now try one for yourself.

Example

Solve x² + 6x – 2 = 0

From the previous examples, we know that x² + 6x – 2 = 0 can be written as (x + 3)² – 11 = 0

So, to solve the equation, take the square root of both sides. So (x + 3)² = 11

x + 3 = + √11

or x + 3 = – √11

x = – 3 + √11

or x = – 3 –  √11

x = – 3 + 3.317 or x = – 3 – 3.317 (√11 is 3.317)

x = 0.317 (3 s.f) or x = – 6.317 (3 s.f)

Example

Rewrite 2x² + 20x + 3

Rewrite to get x² on its own.

2( x² + 10x ) + 3

The coefficient of x is 10. Divide 10 by 2, and square to get 25.

= 2 ( ( x + 5)² – 25) + 3

= 2 (x + 5)² – 50 + 3

= 2 (x + 5)² – 47

Now use the previous example to solve 2x² + 20x + 3 = 0

From the previous example, we know that 2x² + 20x + 3 can be rewritten as:

2 (x + 5)² – 47

Therefore, we can rewrite the equation as:

2(x + 5 )² – 47 = 0

2(x + 5 )² = 47

(x + 5 )² = 23.5 (dividing both sides by 2)

Take the square root of both sides.

x + 5 = √23.5

or x + 5 = – √23.5

x = – 5 + √23.5

or x = – 5 – √23.5

x = – 5 + √23.5

or x = – 5 – √23.5

x = – 0.152 (3 s.f) or x = – 9.85 (3 s.f)

Exercise

1. x² + 4x – 5

2. x² + 6x – 7

3. (a + 4)²

4. (3x + y)²

5. 49m² – n²