1. **Problem Statement:** Solve the equations in parts b, c, d, e, and factorise and simplify in parts f and g as requested. --- ### b) Solve the following equations WITHOUT using the quadratic formula: **i) $x^2 - 7x + 12 = 0$** - Step 1: Factorise the quadratic. - Step 2: Find two numbers that multiply to 12 and add to -7: -3 and -4. - Step 3: Write as $(x - 3)(x - 4) = 0$. - Step 4: Set each factor to zero: $x - 3 = 0$ or $x - 4 = 0$. - Step 5: Solutions: $x = 3$ or $x = 4$. **ii) $x^2 - 6x = 0$** - Step 1: Factor out $x$: $x(x - 6) = 0$. - Step 2: Set each factor to zero: $x = 0$ or $x - 6 = 0$. - Step 3: Solutions: $x = 0$ or $x = 6$. **iii) $x^2 - 25 = 0$** - Step 1: Recognize difference of squares: $x^2 - 5^2 = 0$. - Step 2: Factor as $(x - 5)(x + 5) = 0$. - Step 3: Solutions: $x = 5$ or $x = -5$. **iv) $4x^2 - 12x + 5 = 0$** - Step 1: Try to factorise. - Step 2: Multiply $4 imes 5 = 20$. - Step 3: Find two numbers that multiply to 20 and add to -12: -10 and -2. - Step 4: Rewrite: $4x^2 - 10x - 2x + 5 = 0$. - Step 5: Factor by grouping: $2x(2x - 5) -1(2x - 5) = 0$. - Step 6: Factor out $(2x - 5)$: $(2x - 5)(2x - 1) = 0$. - Step 7: Solutions: $2x - 5 = 0 \Rightarrow x = \frac{5}{2}$, $2x - 1 = 0 \Rightarrow x = \frac{1}{2}$. **v) $5x^2 - 6x = 0$** - Step 1: Factor out $x$: $x(5x - 6) = 0$. - Step 2: Solutions: $x = 0$ or $5x - 6 = 0 \Rightarrow x = \frac{6}{5}$. **vi) $9x^2 - 16 = 0$** - Step 1: Recognize difference of squares: $(3x)^2 - 4^2 = 0$. - Step 2: Factor: $(3x - 4)(3x + 4) = 0$. - Step 3: Solutions: $x = \frac{4}{3}$ or $x = -\frac{4}{3}$. **vii) $(x + 3)^2 = (x + 1)(2x + 3)$** - Step 1: Expand both sides: - Left: $(x + 3)^2 = x^2 + 6x + 9$. - Right: $(x + 1)(2x + 3) = 2x^2 + 3x + 2x + 3 = 2x^2 + 5x + 3$. - Step 2: Set equation: $x^2 + 6x + 9 = 2x^2 + 5x + 3$. - Step 3: Rearrange: $0 = 2x^2 + 5x + 3 - x^2 - 6x - 9 = x^2 - x - 6$. - Step 4: Factor: $(x - 3)(x + 2) = 0$. - Step 5: Solutions: $x = 3$ or $x = -2$. **viii) $5x^2 - 13x = 6$** - Step 1: Rearrange: $5x^2 - 13x - 6 = 0$. - Step 2: Multiply $5 imes (-6) = -30$. - Step 3: Find two numbers that multiply to -30 and add to -13: -15 and 2. - Step 4: Rewrite: $5x^2 - 15x + 2x - 6 = 0$. - Step 5: Factor by grouping: $5x(x - 3) + 2(x - 3) = 0$. - Step 6: Factor out $(x - 3)$: $(5x + 2)(x - 3) = 0$. - Step 7: Solutions: $x = -\frac{2}{5}$ or $x = 3$. **ix) $4x^2 = 1$** - Step 1: Rearrange: $4x^2 - 1 = 0$. - Step 2: Recognize difference of squares: $(2x)^2 - 1^2 = 0$. - Step 3: Factor: $(2x - 1)(2x + 1) = 0$. - Step 4: Solutions: $x = \frac{1}{2}$ or $x = -\frac{1}{2}$. --- ### c) Solve the following equations, answers correct to 2 decimal places: **i) $x^2 + 2x - 5 = 0$** - Step 1: Use quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with $a=1$, $b=2$, $c=-5$. - Step 2: Calculate discriminant: $2^2 - 4(1)(-5) = 4 + 20 = 24$. - Step 3: Calculate roots: $$x = \frac{-2 \pm \sqrt{24}}{2} = \frac{-2 \pm 2\sqrt{6}}{2} = -1 \pm \sqrt{6}$$ - Step 4: Approximate: - $x_1 = -1 + 2.45 = 1.45$ - $x_2 = -1 - 2.45 = -3.45$ **ii) $3x^2 + 7x - 5 = 0$** - Step 1: $a=3$, $b=7$, $c=-5$. - Step 2: Discriminant: $7^2 - 4(3)(-5) = 49 + 60 = 109$. - Step 3: Roots: $$x = \frac{-7 \pm \sqrt{109}}{6}$$ - Step 4: Approximate: - $x_1 = \frac{-7 + 10.44}{6} = 0.57$ - $x_2 = \frac{-7 - 10.44}{6} = -2.57$ **iii) $2x^2 = 7x - 4$** - Step 1: Rearrange: $2x^2 - 7x + 4 = 0$. - Step 2: $a=2$, $b=-7$, $c=4$. - Step 3: Discriminant: $(-7)^2 - 4(2)(4) = 49 - 32 = 17$. - Step 4: Roots: $$x = \frac{7 \pm \sqrt{17}}{4}$$ - Step 5: Approximate: - $x_1 = \frac{7 + 4.12}{4} = 2.78$ - $x_2 = \frac{7 - 4.12}{4} = 0.72$ --- ### d) Solve the following equations, answers in surd form: **i) $3x^2 - 6x + 2 = 0$** - Step 1: $a=3$, $b=-6$, $c=2$. - Step 2: Discriminant: $(-6)^2 - 4(3)(2) = 36 - 24 = 12$. - Step 3: Roots: $$x = \frac{6 \pm \sqrt{12}}{6} = \frac{6 \pm 2\sqrt{3}}{6} = 1 \pm \frac{\sqrt{3}}{3}$$ **ii) $4x^2 + 3x = 5$** - Step 1: Rearrange: $4x^2 + 3x - 5 = 0$. - Step 2: $a=4$, $b=3$, $c=-5$. - Step 3: Discriminant: $3^2 - 4(4)(-5) = 9 + 80 = 89$. - Step 4: Roots: $$x = \frac{-3 \pm \sqrt{89}}{8}$$ **iii) $2x^2 = 7x - 4$** - Step 1: Rearrange: $2x^2 - 7x + 4 = 0$. - Step 2: $a=2$, $b=-7$, $c=4$. - Step 3: Discriminant: $(-7)^2 - 4(2)(4) = 49 - 32 = 17$. - Step 4: Roots: $$x = \frac{7 \pm \sqrt{17}}{4}$$ --- ### e) Find quadratic equations with given roots: **i) Roots: -4 and 3** - Step 1: Use formula: $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$. - Step 2: Sum: $-4 + 3 = -1$. - Step 3: Product: $-4 \times 3 = -12$. - Step 4: Equation: $x^2 - (-1)x + (-12) = x^2 + x - 12 = 0$. **ii) Roots: $\frac{2}{3}$ and $\frac{1}{5}$** - Step 1: Sum: $\frac{2}{3} + \frac{1}{5} = \frac{10}{15} + \frac{3}{15} = \frac{13}{15}$. - Step 2: Product: $\frac{2}{3} \times \frac{1}{5} = \frac{2}{15}$. - Step 3: Equation: $$x^2 - \frac{13}{15}x + \frac{2}{15} = 0$$ - Step 4: Multiply all terms by 15 to clear denominators: $$15x^2 - 13x + 2 = 0$$ --- ### f) Factorise the following: **i) $p^3 - 25p$** - Step 1: Factor out $p$: $p(p^2 - 25)$. - Step 2: Recognize difference of squares: $p(p - 5)(p + 5)$. **ii) $4x^2 - 8x - 12$** - Step 1: Factor out 4: $4(x^2 - 2x - 3)$. - Step 2: Factor quadratic: $4(x - 3)(x + 1)$. **iii) $3p^2 - 3p - 18$** - Step 1: Factor out 3: $3(p^2 - p - 6)$. - Step 2: Factor quadratic: $3(p - 3)(p + 2)$. **iv) $8m^3 - 50m$** - Step 1: Factor out $2m$: $2m(4m^2 - 25)$. - Step 2: Recognize difference of squares: $2m(2m - 5)(2m + 5)$. --- ### g) Simplify the following: **i) $\frac{8a + 8b}{a + b}$** - Step 1: Factor numerator: $8(a + b)$. - Step 2: Simplify: $$\frac{8\cancel{(a + b)}}{\cancel{(a + b)}} = 8$$ **ii) $\frac{a - 2}{a^2 + 5a - 14}$** - Step 1: Factor denominator: $(a + 7)(a - 2)$. - Step 2: Simplify: $$\frac{a - 2}{(a + 7)(a - 2)} = \frac{\cancel{a - 2}}{(a + 7)\cancel{(a - 2)}} = \frac{1}{a + 7}$$ **iii) $\frac{x^2 + x - 30}{x - 5}$** - Step 1: Factor numerator: $(x + 6)(x - 5)$. - Step 2: Simplify: $$\frac{(x + 6)\cancel{(x - 5)}}{\cancel{x - 5}} = x + 6$$ **iv) $\frac{ab - ac}{b - c}$** - Step 1: Factor numerator: $a(b - c)$. - Step 2: Simplify: $$\frac{a\cancel{(b - c)}}{\cancel{b - c}} = a$$ **vii) $\frac{m - n}{2m - 3n}$** - Step 1: No common factors; expression is simplified. **viii) $\frac{p^2 - q^2}{p^2 + 2pq + q^2}$** - Step 1: Factor numerator: $(p - q)(p + q)$. - Step 2: Factor denominator: $(p + q)^2$. - Step 3: Simplify: $$\frac{(p - q)(p + q)}{(p + q)^2} = \frac{p - q}{p + q}$$ --- Final answers are shown step-by-step with explanations and simplifications as requested.