2. Solve each equation by rewriting it in factored form and using the zero product property. Show your reasoning. a. \( x^{2}+8 x+15=0 \) b. \( x^{2}-8 x+12=5 \) c. \( x^{2}-10 x-11=0 \) d. \( 49-x^{2}=0 \) e. \( (x+4)(x+5)-30=0 \)

Sure, let's solve each equation by factoring and applying the Zero Product Property. The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. --- ### **a. \( x^{2} + 8x + 15 = 0 \)** **Step 1: Factor the quadratic equation.** We need to find two numbers that multiply to **15** (the constant term) and add up to **8** (the coefficient of \( x \)). - The numbers **3** and **5** satisfy this because: - \( 3 \times 5 = 15 \) - \( 3 + 5 = 8 \) **Step 2: Write the factored form.** \[ x^{2} + 8x + 15 = (x + 3)(x + 5) = 0 \] **Step 3: Apply the Zero Product Property.** \[ x + 3 = 0 \quad \text{or} \quad x + 5 = 0 \] **Step 4: Solve for \( x \).** \[ x = -3 \quad \text{or} \quad x = -5 \] **Solution:** \( x = -3 \) and \( x = -5 \) --- ### **b. \( x^{2} - 8x + 12 = 5 \)** **Step 1: Bring the equation to standard quadratic form.** Subtract **5** from both sides: \[ x^{2} - 8x + 12 - 5 = 0 \\ x^{2} - 8x + 7 = 0 \] **Step 2: Factor the quadratic equation.** Find two numbers that multiply to **7** and add up to **-8**. - The numbers **-1** and **-7** work because: - \( (-1) \times (-7) = 7 \) - \( -1 + (-7) = -8 \) **Step 3: Write the factored form.** \[ x^{2} - 8x + 7 = (x - 1)(x - 7) = 0 \] **Step 4: Apply the Zero Product Property.** \[ x - 1 = 0 \quad \text{or} \quad x - 7 = 0 \] **Step 5: Solve for \( x \).** \[ x = 1 \quad \text{or} \quad x = 7 \] **Solution:** \( x = 1 \) and \( x = 7 \) --- ### **c. \( x^{2} - 10x - 11 = 0 \)** **Step 1: Factor the quadratic equation.** We need two numbers that multiply to **-11** and add up to **-10**. - The numbers **-11** and **1** work because: - \( -11 \times 1 = -11 \) - \( -11 + 1 = -10 \) **Step 2: Write the factored form.** \[ x^{2} - 10x - 11 = (x - 11)(x + 1) = 0 \] **Step 3: Apply the Zero Product Property.** \[ x - 11 = 0 \quad \text{or} \quad x + 1 = 0 \] **Step 4: Solve for \( x \).** \[ x = 11 \quad \text{or} \quad x = -1 \] **Solution:** \( x = 11 \) and \( x = -1 \) --- ### **d. \( 49 - x^{2} = 0 \)** **Step 1: Rewrite the equation.** \[ 49 - x^{2} = 0 \\ -x^{2} + 49 = 0 \\ x^{2} = 49 \] **Step 2: Factor the difference of squares.** \[ x^{2} - 49 = 0 \\ (x - 7)(x + 7) = 0 \] **Step 3: Apply the Zero Product Property.** \[ x - 7 = 0 \quad \text{or} \quad x + 7 = 0 \] **Step 4: Solve for \( x \).** \[ x = 7 \quad \text{or} \quad x = -7 \] **Solution:** \( x = 7 \) and \( x = -7 \) --- ### **e. \( (x + 4)(x + 5) - 30 = 0 \)** **Step 1: Expand the expression and bring to standard form.** \[ (x + 4)(x + 5) - 30 = 0 \\ x^{2} + 5x + 4x + 20 - 30 = 0 \\ x^{2} + 9x - 10 = 0 \] **Step 2: Factor the quadratic equation.** Find two numbers that multiply to **-10** and add up to **9**. - The numbers **10** and **-1** work because: - \( 10 \times (-1) = -10 \) - \( 10 + (-1) = 9 \) **Step 3: Write the factored form.** \[ x^{2} + 9x - 10 = (x + 10)(x - 1) = 0 \] **Step 4: Apply the Zero Product Property.** \[ x + 10 = 0 \quad \text{or} \quad x - 1 = 0 \] **Step 5: Solve for \( x \).** \[ x = -10 \quad \text{or} \quad x = 1 \] **Solution:** \( x = -10 \) and \( x = 1 \) --- Each equation has been factored, and the solutions have been found by applying the Zero Product Property.