Quadratic Factorising Fa9425

1. **Problem:** Solve the quadratic equation $x^2 - 7x + 12 = 0$ by factorising. 2. **Formula and rules:** To solve quadratic equations by factorising, express the quadratic in the form $ax^2 + bx + c = 0$ and factor it into two binomials $(mx + n)(px + q) = 0$. Then set each factor equal to zero and solve for $x$. 3. **Step-by-step solution for 1a:** - Factor $x^2 - 7x + 12$. - Find two numbers that multiply to $12$ and add to $-7$: these are $-3$ and $-4$. - So, $x^2 - 7x + 12 = (x - 3)(x - 4) = 0$. - Set each factor to zero: $$x - 3 = 0 \Rightarrow x = 3$$ $$x - 4 = 0 \Rightarrow x = 4$$ 4. **Answer:** $x = 3$ or $x = 4$ --- 1. **Problem:** Solve $x^2 + 5x - 6 = 0$ by factorising. 2. **Step-by-step:** - Find two numbers that multiply to $-6$ and add to $5$: $6$ and $-1$. - Factor: $(x + 6)(x - 1) = 0$. - Solve: $$x + 6 = 0 \Rightarrow x = -6$$ $$x - 1 = 0 \Rightarrow x = 1$$ 3. **Answer:** $x = -6$ or $x = 1$ --- 1. **Problem:** Solve $x^2 - 25 = 0$. 2. **Step-by-step:** - Recognize difference of squares: $x^2 - 25 = (x - 5)(x + 5) = 0$. - Solve: $$x - 5 = 0 \Rightarrow x = 5$$ $$x + 5 = 0 \Rightarrow x = -5$$ 3. **Answer:** $x = 5$ or $x = -5$ --- 1. **Problem:** Solve $x^2 + 11x + 30 = 0$. 2. **Step-by-step:** - Find two numbers that multiply to $30$ and add to $11$: $5$ and $6$. - Factor: $(x + 5)(x + 6) = 0$. - Solve: $$x + 5 = 0 \Rightarrow x = -5$$ $$x + 6 = 0 \Rightarrow x = -6$$ 3. **Answer:** $x = -5$ or $x = -6$ --- 1. **Problem:** Solve $x^2 - 2x - 15 = 0$. 2. **Step-by-step:** - Find two numbers that multiply to $-15$ and add to $-2$: $3$ and $-5$. - Factor: $(x + 3)(x - 5) = 0$. - Solve: $$x + 3 = 0 \Rightarrow x = -3$$ $$x - 5 = 0 \Rightarrow x = 5$$ 3. **Answer:** $x = -3$ or $x = 5$ --- 1. **Problem:** Solve $x^2 + 7x + 10 = 0$. 2. **Step-by-step:** - Find two numbers that multiply to $10$ and add to $7$: $5$ and $2$. - Factor: $(x + 5)(x + 2) = 0$. - Solve: $$x + 5 = 0 \Rightarrow x = -5$$ $$x + 2 = 0 \Rightarrow x = -2$$ 3. **Answer:** $x = -5$ or $x = -2$ --- 1. **Problem:** Solve $x^2 - 8x + 12 = 0$. 2. **Step-by-step:** - Find two numbers that multiply to $12$ and add to $-8$: $-6$ and $-2$. - Factor: $(x - 6)(x - 2) = 0$. - Solve: $$x - 6 = 0 \Rightarrow x = 6$$ $$x - 2 = 0 \Rightarrow x = 2$$ 3. **Answer:** $x = 6$ or $x = 2$ --- 1. **Problem:** Solve $x^2 + 15x - 16 = 0$. 2. **Step-by-step:** - Find two numbers that multiply to $-16$ and add to $15$: $16$ and $-1$. - Factor: $(x + 16)(x - 1) = 0$. - Solve: $$x + 16 = 0 \Rightarrow x = -16$$ $$x - 1 = 0 \Rightarrow x = 1$$ 3. **Answer:** $x = -16$ or $x = 1$ --- 1. **Problem:** Solve $x^2 + x - 12 = 0$. 2. **Step-by-step:** - Find two numbers that multiply to $-12$ and add to $1$: $4$ and $-3$. - Factor: $(x + 4)(x - 3) = 0$. - Solve: $$x + 4 = 0 \Rightarrow x = -4$$ $$x - 3 = 0 \Rightarrow x = 3$$ 3. **Answer:** $x = -4$ or $x = 3$ --- 1. **Problem:** Solve $x^2 - x - 20 = 0$. 2. **Step-by-step:** - Find two numbers that multiply to $-20$ and add to $-1$: $-5$ and $4$. - Factor: $(x - 5)(x + 4) = 0$. - Solve: $$x - 5 = 0 \Rightarrow x = 5$$ $$x + 4 = 0 \Rightarrow x = -4$$ 3. **Answer:** $x = 5$ or $x = -4$ --- 1. **Problem:** Solve $x^2 + 5x - 24 = 0$. 2. **Step-by-step:** - Find two numbers that multiply to $-24$ and add to $5$: $8$ and $-3$. - Factor: $(x + 8)(x - 3) = 0$. - Solve: $$x + 8 = 0 \Rightarrow x = -8$$ $$x - 3 = 0 \Rightarrow x = 3$$ 3. **Answer:** $x = -8$ or $x = 3$ --- 1. **Problem:** Solve $x^2 - 49 = 0$. 2. **Step-by-step:** - Recognize difference of squares: $x^2 - 49 = (x - 7)(x + 7) = 0$. - Solve: $$x - 7 = 0 \Rightarrow x = 7$$ $$x + 7 = 0 \Rightarrow x = -7$$ 3. **Answer:** $x = 7$ or $x = -7$ --- 1. **Problem:** Solve $40 - 3x - x^2 = 0$. 2. **Step-by-step:** - Rewrite as $-x^2 - 3x + 40 = 0$ or $x^2 + 3x - 40 = 0$ by multiplying both sides by $-1$. - Factor $x^2 + 3x - 40$. - Find two numbers that multiply to $-40$ and add to $3$: $8$ and $-5$. - Factor: $(x + 8)(x - 5) = 0$. - Solve: $$x + 8 = 0 \Rightarrow x = -8$$ $$x - 5 = 0 \Rightarrow x = 5$$ 3. **Answer:** $x = -8$ or $x = 5$ --- 1. **Problem:** Solve $7 + 6x - x^2 = 0$. 2. **Step-by-step:** - Rewrite as $-x^2 + 6x + 7 = 0$ or $x^2 - 6x - 7 = 0$ by multiplying both sides by $-1$. - Find two numbers that multiply to $-7$ and add to $-6$: $-7$ and $1$. - Factor: $(x - 7)(x + 1) = 0$. - Solve: $$x - 7 = 0 \Rightarrow x = 7$$ $$x + 1 = 0 \Rightarrow x = -1$$ 3. **Answer:** $x = 7$ or $x = -1$ --- 1. **Problem:** Solve $8 + 2x - x^2 = 0$. 2. **Step-by-step:** - Rewrite as $-x^2 + 2x + 8 = 0$ or $x^2 - 2x - 8 = 0$ by multiplying both sides by $-1$. - Find two numbers that multiply to $-8$ and add to $-2$: $-4$ and $2$. - Factor: $(x - 4)(x + 2) = 0$. - Solve: $$x - 4 = 0 \Rightarrow x = 4$$ $$x + 2 = 0 \Rightarrow x = -2$$ 3. **Answer:** $x = 4$ or $x = -2$