Questions: Use factoring to solve the quadratic equation. Check by solving x^2 - 4x - 21 = 0 The solution set is .

Solution Steps

To solve the quadratic equation \(x^2 - 4x - 21 = 0\) by factoring, we need to find two numbers that multiply to -21 and add up to -4. Once we find these numbers, we can factor the quadratic equation and solve for \(x\). Finally, we can check our solutions by substituting them back into the original equation.

We start with the quadratic equation: \[ x^2 - 4x - 21 = 0 \]

We need to factor the quadratic equation. We look for two numbers that multiply to \(-21\) and add up to \(-4\). These numbers are \(-7\) and \(3\). Thus, we can factor the equation as: \[ (x - 7)(x + 3) = 0 \]

To find the solutions, we set each factor equal to zero and solve for \(x\): \[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \] \[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]

We substitute \(x = 7\) and \(x = -3\) back into the original equation to verify: For \(x = 7\): \[ 7^2 - 4(7) - 21 = 49 - 28 - 21 = 0 \] For \(x = -3\): \[ (-3)^2 - 4(-3) - 21 = 9 + 12 - 21 = 0 \] Both solutions satisfy the original equation.

Final Answer