Solve the equation by the zero-factor property 6) \( x^{2}+2 x-15=0 \)

To solve the quadratic equation \( x^{2} + 2x - 15 = 0 \) using the **Zero-Factor Property**, follow these steps: ### Step 1: Factor the Quadratic Equation The Zero-Factor Property states that if the product of two factors is zero, then at least one of the factors must be zero. So, we need to factor the quadratic expression \( x^{2} + 2x - 15 \) into two binomials. **Find two numbers** that: 1. **Multiply to give the constant term (-15).** 2. **Add up to give the coefficient of the middle term (2).** Let's list the pairs of integers that multiply to -15: - \( 1 \) and \( -15 \) (since \( 1 \times (-15) = -15 \)) - \( -1 \) and \( 15 \) (since \( -1 \times 15 = -15 \)) - \( 3 \) and \( -5 \) (since \( 3 \times (-5) = -15 \)) - \( -3 \) and \( 5 \) (since \( -3 \times 5 = -15 \)) Now, identify which pair adds up to **2**: - \( 3 + (-5) = -2 \) ❌ - \( -3 + 5 = 2 \) ✅ So, the numbers are **-3** and **5**. ### Step 2: Write the Factored Form Using the numbers found, factor the quadratic as follows: \[ x^{2} + 2x - 15 = (x - 3)(x + 5) = 0 \] ### Step 3: Apply the Zero-Factor Property Set each factor equal to zero and solve for \( x \): \[ x - 3 = 0 \quad \text{or} \quad x + 5 = 0 \] **Solving these equations:** 1. \( x - 3 = 0 \) \( \Rightarrow x = 3 \) 2. \( x + 5 = 0 \) \( \Rightarrow x = -5 \) ### **Solutions** \[ x = 3 \quad \text{and} \quad x = -5 \] These are the values of \( x \) that satisfy the original equation \( x^{2} + 2x - 15 = 0 \).