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Expanding and factorizing quadratic expressions

1. Expanding two brackets

2. Squaring expressions

3. The difference between two squares

4. Factorizing expressions

5. Quadratic expressions

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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.

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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.

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Matching quadratic expressions 1

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Matching quadratic expressions 2

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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

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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

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Squaring expressions

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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 (a + b)(a – b) = a2 – b2

This is the difference between two squares.

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Matching the difference between two squares

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Factorizing expressions

Writing 5x + 10 as 5(x + 2) is called factorizing the expression.

3x + x2 = x(3 + x)2p + 6p2 – 4p3

= 2p(1 + 3p – 2p2)

The highest common factor of 3x and x2 is x.

(3x + x2) ÷ x = 3 + x

The highest common factor of 2p, 6p2 and 4p3 is 2p.

(2p + 6p2 – 4p3) ÷ 2p

= 1 + 3p – 2p2

Factorize 3x + x2 Factorize 2p + 6p2 – 4p3

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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)

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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.

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Factorizing quadratic expressions

Quadratic expressions of the form x2 + bx + c can be factorized if they can be written using brackets as

(x + d)(x + e)

where d and e are integers.

If we expand (x + d)(x + e) we have,

(x + d)(x + e) = x2 + dx + ex + de

= x2 + (d + e)x + de

Comparing this to x2 + bx + c we can see that:

The sum of d and e must be equal to b, the coefficient of x.

The product of d and e must be equal to c, the constant term.

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Factorizing quadratic expressions 1

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Matching quadratic expressions 1

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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

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Factorizing quadratic expressions 2

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Matching quadratic expressions 2

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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)

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Factorizing the difference between two squares

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Matching the difference between two squares