Question

Factorise the following:
a) x^2 + 10x + 16
b) x^2 + 7x + 12
c) x^2 + 13x + 12

Answer 1

Explanation:

a)

`x^2+10x+16=\\x^2+2x+8x+16=\\x(x+2)+8(x+2)=\\(x+2)(x+8)`

b)

`x^2+7x+12=\\x^2+3x+4x+12=\\x(x+3)+4(x+3)=\\(x+3)(x+4)\\`

c)

`x^2+13x+12=\\x^2+x+12x+12=\\x(x+1)+12(x+1)=\\(x+1)(x+12)`

Answer 2

`\textsf{(a)} \quad (x+8)(x+2)`

`\textsf{(b)} \quad (x+4)(x+3)`

`\textsf{(c)} \quad (x+12)(x+1)`

Explanation:

Factoring quadratics

• To factor a quadratic in the form
  `ax^2+bx+c` find two numbers that multiply to
  `ac` and sum to
  `b`.
• Rewrite
  `b` as the sum of these two numbers.
• Factor the first two terms and the last two terms separately.
• Factor out the common term.

Part (a)

`\textsf{Given}: \quad x^2+10x+16`

Two numbers that multiply to 16 and sum to 10 are: 2 and 8.

`\implies x^2+2x+8x+16`

Factor the first two terms and the last two terms separately:

`\implies x(x+2)+8(x+2)`

Factor out the common term (x + 2):

`\implies (x+8)(x+2)`

Part (b)

`\textsf{Given}: \quad x^2+7x+12`

Two numbers that multiply to 12 and sum to 7 are: 3 and 4.

`\implies x^2+3x+4x+12`

Factor the first two terms and the last two terms separately:

`\implies x(x+3)+4(x+3)`

Factor out the common term (x + 3):

`\implies (x+4)(x+3)`

Part (c)

`\textsf{Given}: \quad x^2+13x+12`

Two numbers that multiply to 12 and sum to 13 are: 1 and 12.

`\implies x^2+x+12x+12`

Factor the first two terms and the last two terms separately:

`\implies x(x+1)+12(x+1)`

Factor out the common term (x + 1):

`\implies (x+12)(x+1)`