Solve the equation by factoring. \( 8 x^{2}+10 x-7=0 \) Rewrite the equation in factored form. \( \square=0 \) (Factor completely.) The solution set is \( \{\square\} \) (Use a comma to separate answers as needed. Type each solution only once.)

To solve the quadratic equation \( 8x^{2} + 10x - 7 = 0 \) by factoring, follow these steps: ### Step 1: Factor the Equation 1. **Identify coefficients**: - \( a = 8 \) - \( b = 10 \) - \( c = -7 \) 2. **Find two numbers that multiply to \( a \times c = 8 \times (-7) = -56 \) and add up to \( b = 10 \)**: - The numbers are **14** and **-4** because: - \( 14 \times (-4) = -56 \) - \( 14 + (-4) = 10 \) 3. **Rewrite the middle term using these numbers**: \[ 8x^{2} + 14x - 4x - 7 = 0 \] 4. **Factor by grouping**: \[ (8x^{2} + 14x) + (-4x - 7) = 0 \\ 2x(4x + 7) - 1(4x + 7) = 0 \] 5. **Factor out the common binomial**: \[ (2x - 1)(4x + 7) = 0 \] So, the **factored form** of the equation is: \[ (2x - 1)(4x + 7) = 0 \] ### Step 2: Find the Solution Set Set each factor equal to zero and solve for \( x \): 1. \( 2x - 1 = 0 \) \[ 2x = 1 \\ x = \frac{1}{2} \] 2. \( 4x + 7 = 0 \) \[ 4x = -7 \\ x = -\frac{7}{4} \] Therefore, the **solution set** is: \[ \left\{ -\frac{7}{4}, \frac{1}{2} \right\} \] ### Final Answer - **Factored Form**: \[ (2x - 1)(4x + 7) = 0 \] - **Solution Set**: \[ \left\{ -\frac{7}{4}, \frac{1}{2} \right\} \]