Use factoring to solve the quadratic equation. Check by substitution or by using a graphing utility and identifying \( x \)-intercepts. \[ x^{2}-x-12=0 \]

To solve the quadratic equation \( x^{2} - x - 12 = 0 \) by factoring and verify the solutions, follow these steps: ### 1. Factoring the Quadratic Equation We need to express the quadratic equation in the form: \[ x^{2} - x - 12 = (x + a)(x + b) = 0 \] where \( a \) and \( b \) are numbers that satisfy: \[ a + b = -1 \quad \text{and} \quad a \times b = -12 \] **Finding \( a \) and \( b \):** - List the pairs of factors of \(-12\): - \( 1 \) and \(-12\) → Sum: \( 1 + (-12) = -11 \) - \(-1\) and \(12\) → Sum: \(-1 + 12 = 11 \) - \( 3 \) and \(-4\) → Sum: \( 3 + (-4) = -1 \) - \(-3\) and \(4\) → Sum: \(-3 + 4 = 1 \) The pair that sums to \(-1\) is \( 3 \) and \(-4\). **Factored Form:** \[ x^{2} - x - 12 = (x + 3)(x - 4) = 0 \] ### 2. Solving for \( x \) Set each factor equal to zero: 1. \( x + 3 = 0 \) \(\Rightarrow x = -3 \) 2. \( x - 4 = 0 \) \(\Rightarrow x = 4 \) **Solutions:** \( x = -3 \) and \( x = 4 \) ### 3. Checking the Solutions by Substitution **For \( x = -3 \):** \[ (-3)^2 - (-3) - 12 = 9 + 3 - 12 = 0 \] **For \( x = 4 \):** \[ (4)^2 - 4 - 12 = 16 - 4 - 12 = 0 \] Both solutions satisfy the original equation. ### 4. Verifying with a Graphing Utility By graphing the quadratic function \( y = x^{2} - x - 12 \), the \( x \)-intercepts (where \( y = 0 \)) occur at the solutions \( x = -3 \) and \( x = 4 \). This visual confirmation aligns with our algebraic solutions. **Graphical Representation:**  *The graph intersects the \( x \)-axis at \( x = -3 \) and \( x = 4 \).* ### **Final Answer:** The solutions to the equation \( x^{2} - x - 12 = 0 \) are: \[ x = -3 \quad \text{and} \quad x = 4 \]