Question

Factorise by substituting terms:

(a² – 2a)² – 18(a² – 2a) + 45

Answer 1

The given expression is (a² – 2a)² – 18(a² – 2a) + 45.

Step 1: Substitute a term

Let x = a² – 2a.

The expression is transformed into a quadratic equation in terms of x:

x² – 18x + 45

Step 2: Factorise the quadratic expression

The quadratic expression on x² – 18x + 45 is factorised by finding two numbers that multiply to 45 and add up to –18.

These numbers are –3 and –15.

The expression is rewritten as x² – 3x – 15x + 45

Common factors are taken out x(x – 3) – 15(x – 3)

The expression is factorised as (x – 3) (x – 15)

Step 3: Substitute back the original term

The original term a² – 2a is substituted back for x:

(a² – 2a – 3) (a² – 2a – 15)

Step 4: Factorise the resulting quadratic expressions

Each of the two quadratic expressions is factorised further.

For the first expression, a² – 2a – 3:

Two numbers that multiply to –3 and add up to –2 are – 3 and 1.

The expression is rewritten as a² – 3a + a – 3

Common factors are taken out a(a – 3) + 1(a – 3)

The expression is factorised as (a – 3) (a + 1)

For the second expression, a² – 2a – 15:

Two numbers that multiply to –15 and add up to –2 are –5 and 3.

The expression is rewritten as a² – 5a + 3a – 15

Common factors are taken out a(a – 5) + 3(a – 5)

The expression is factorised as (a – 5) (a + 3)

The factorised form of the expression (a² – 2a)² – 18(a² – 2a) + 45 is (a – 3) (a + 1) (a – 5) (a + 3).