High School

**Q1.**

(a)
(i) Factorise [tex]x^2 - 12x + 27[/tex].

(ii) Solve the equation [tex]x^2 - 12x + 27 = 0[/tex].

(b) Factorise [tex]y^2 - 100[/tex].

(Total for Question 1 is 4 marks)

**Q2.**

Solve, by factorising, the equation [tex]8x^2 - 30x - 27 = 0[/tex].

Answer :

Let's go through each part of the problem step-by-step.

Q1(a)(i): Factorise [tex]\(x^2 - 12x + 27\)[/tex]

To factor the quadratic expression [tex]\(x^2 - 12x + 27\)[/tex], we look for two numbers that multiply to 27 (the constant term) and add to -12 (the coefficient of x).

The numbers -3 and -9 satisfy these conditions because:
- [tex]\((-3) \times (-9) = 27\)[/tex]
- [tex]\((-3) + (-9) = -12\)[/tex]

Therefore, we can factor the expression as:
[tex]\[
x^2 - 12x + 27 = (x - 3)(x - 9)
\][/tex]

Q1(a)(ii): Solve the equation [tex]\(x^2 - 12x + 27 = 0\)[/tex]

We already factored the quadratic:
[tex]\[
(x - 3)(x - 9) = 0
\][/tex]

To solve, set each factor to zero:
1. [tex]\(x - 3 = 0\)[/tex] gives [tex]\(x = 3\)[/tex]
2. [tex]\(x - 9 = 0\)[/tex] gives [tex]\(x = 9\)[/tex]

The solutions to the equation are [tex]\(x = 3\)[/tex] and [tex]\(x = 9\)[/tex].

Q1(b): Factorise [tex]\(y^2 - 100\)[/tex]

The expression [tex]\(y^2 - 100\)[/tex] is a difference of squares. It can be factored using the formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].

Here, [tex]\(y^2 - 100\)[/tex] is:
[tex]\[
y^2 - 10^2 = (y - 10)(y + 10)
\][/tex]

Q2: Solve, by factorising, the equation [tex]\(8x^2 - 30x - 27 = 0\)[/tex]

We need to factor this quadratic expression. The factors are [tex]\((2x - 9)\)[/tex] and [tex]\((4x + 3)\)[/tex].

[tex]\[
(2x - 9)(4x + 3) = 0
\][/tex]

To find the solutions, set each factor to zero:
1. [tex]\(2x - 9 = 0\)[/tex] gives [tex]\(2x = 9\)[/tex], so [tex]\(x = \frac{9}{2}\)[/tex]
2. [tex]\(4x + 3 = 0\)[/tex] gives [tex]\(4x = -3\)[/tex], so [tex]\(x = -\frac{3}{4}\)[/tex]

Thus, the solutions to the equation are [tex]\(x = \frac{9}{2}\)[/tex] and [tex]\(x = -\frac{3}{4}\)[/tex].

I hope this helps clarify the solutions! Let me know if you have any questions.