Stage 8 Chapter 18

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Quadratics

Stage 8 Chapter 18


Objectives

• Multiply expressions of the form (x+3)(x-7) and simplify the resulting expression;

• factorise quadratic expressions including the difference of two squares


You should already know

• How to collect together simple algebraic terms• Expand single brackets• Take out common factors


Expanding two brackets

Look at this algebraic expression:

(3 + t)(4 – 2t)

This means (3 + t) × (4 – 2t), but we do not usually write × in algebra.To expand or multiply out this expression we multiply every term in the second bracket by every term in the first bracket.

(3 + t)(4 – 2t) = 3(4 – 2t) + t(4 – 2t)

= 12 – 6t + 4t – 2t2

= 12 – 2t – 2t2

This is a quadratic

expression.


Complete the activityUsing the grid method to expand brackets


Expanding two brackets

With practice we can expand the product of two linear expressions in fewer steps. For example,

(x – 5)(x + 2) = x2 + 2x – 5x – 10

= x2 – 3x – 10

Notice that –3 is the sum of –5 and 2 …

… and that –10 is the product of –5 and 2.


Complete the activityMatching quadratic expressions 1




Squaring expressions

Expand and simplify: (2 – 3a)2

We can write this as,

(2 – 3a)2 = (2 – 3a)(2 – 3a)

Expanding,

(2 – 3a)(2 – 3a) = 2(2 – 3a) – 3a(2 – 3a)

= 4 – 6a – 6a + 9a2

= 4 – 12a + 9a2


Squaring expressions

In general,

(a + b)2 = a2 + 2ab + b2

The first term squared …

… plus 2 × the product of the two terms …

… plus the second term squared.

For example,

(3m + 2n)2 = 9m2 + 12mn + 4n2


Complete the activitySquaring expressions


The difference between two squares

Expand and simplify (2a + 7)(2a – 7)

Expanding,

(2a + 7)(2a – 7) = 2a(2a – 7) + 7(2a – 7)

= 4a2 – 14a + 14a – 49

= 4a2 – 49

When we simplify, the two middle terms cancel out.

In general,

(a + b)(a – b) = a2 – b2

This is the difference between two squares.


Complete the activityThe difference between two squares


Complete the activityMatching the difference between two squares


Quadratic expressions

A quadratic expression is an expression in which the highest power of the variable is 2. For example,

x2 – 2, w2 + 3w + 1, 4 – 5g2 ,t2

2The general form of a quadratic expression in x is:

x is a variable.

a is a fixed number and is the coefficient of x2.

b is a fixed number and is the coefficient of x.

c is a fixed number and is a constant term.

ax2 + bx + c (where a = 0)


Factorizing expressions

Remember: factorizing an expression is the opposite of expanding it.

Expanding or multiplying out

FactorizingOften:When we expand an expression we remove the brackets.

(a + 1)(a + 2) a2 + 3a + 2

When we factorize an expression we write it with brackets.


Factorizing quadratic expressions

Factorise x² +7x + 12

This will factorise into two brackets with x as the first term in each

x² +7x + 12 = (x )(x )As both the signs are positive, both the numbers will be positive

You need to find two numbers that multiply together to give 12 and add together to give 7

These will be +3 and +4

So x² +7x + 12 = (x + 3)(x + 4) or x² +7x + 12 = (x + 4)(x + 3)


Complete the activityFactorizing quadratic expressions 1




Factorizing quadratic expressions

Quadratic expressions of the form ax2 + bx + c can be factorized if they can be written using brackets as

(dx + e)(fx + g)

where d, e, f and g are integers.

If we expand (dx + e)(fx + g)we have,

(dx + e)(fx + g)= dfx2 + dgx + efx + eg

= dfx2 + (dg + ef)x + eg

Comparing this to ax2 + bx + c we can see that we must choose d, e, f and g such that: a = df,

b = (dg + ef)

c = eg


Complete the activityFactorizing quadratic expressions 2


Matching quadratic expressions 2


Factorizing the difference between two squares

A quadratic expression in the form

x2 – a2

is called the difference between two squares.

The difference between two squares can be factorized as follows:

x2 – a2 = (x + a)(x – a)

For example,

9x2 – 16 = (3x + 4)(3x – 4)

25a2 – 1 = (5a + 1)(5a – 1)

m4 – 49n2 = (m2 + 7n)(m2 – 7n)


Factorizing the difference between two squares


Matching the difference between two squares


Key idas

• When multiplying two brackets, multiply every term in the first bracket by every term in the second

• To factorise x²+ax+b: if b is positive find two numbers that multiply to give b and add up to a

• To factorise x²+ax+b: if b is negative find two numbers that multiply to give b and have a difference of a

• The difference of two squares factorises x²- a² = (x+a)(x-a)

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Stage 8 Chapter 18