Solve for \( x \) by first completing the squere

Certainly! Completing the square is a useful method for solving quadratic equations of the form: \[ ax^2 + bx + c = 0 \] Let's walk through the process step-by-step with a specific example. Suppose we want to solve the quadratic equation: \[ x^2 + 6x + 5 = 0 \] **Step 1: Move the Constant Term to the Other Side** First, subtract the constant term from both sides to set up the equation for completing the square. \[ x^2 + 6x = -5 \] **Step 2: Complete the Square** To complete the square, take half of the coefficient of \( x \), square it, and add it to both sides of the equation. 1. **Find half of the coefficient of \( x \):** \[ \frac{6}{2} = 3 \] 2. **Square this value:** \[ 3^2 = 9 \] 3. **Add this square to both sides:** \[ x^2 + 6x + 9 = -5 + 9 \] **Step 3: Simplify Both Sides** Now, simplify both sides of the equation. \[ x^2 + 6x + 9 = 4 \] The left side of the equation is now a perfect square trinomial. **Step 4: Write the Perfect Square Trinomial as a Binomial Squared** \[ (x + 3)^2 = 4 \] **Step 5: Solve for \( x \)** Take the square root of both sides to solve for \( x \). \[ x + 3 = \pm \sqrt{4} \] \[ x + 3 = \pm 2 \] Now, isolate \( x \): 1. **First solution:** \[ x + 3 = 2 \] \[ x = 2 - 3 \] \[ x = -1 \] 2. **Second solution:** \[ x + 3 = -2 \] \[ x = -2 - 3 \] \[ x = -5 \] **Step 6: Write the Solutions** The solutions to the equation \( x^2 + 6x + 5 = 0 \) are: \[ x = -1 \quad \text{and} \quad x = -5 \] **Verification** To ensure that these solutions are correct, substitute them back into the original equation. 1. **For \( x = -1 \):** \[ (-1)^2 + 6(-1) + 5 = 1 - 6 + 5 = 0 \] 2. **For \( x = -5 \):** \[ (-5)^2 + 6(-5) + 5 = 25 - 30 + 5 = 0 \] Both solutions satisfy the original equation. --- **General Steps to Complete the Square:** If you have a general quadratic equation: \[ ax^2 + bx + c = 0 \] Follow these steps: 1. **Ensure the coefficient of \( x^2 \) is 1.** If not, divide the entire equation by \( a \). 2. **Move the constant term to the other side of the equation.** 3. **Take half of the coefficient of \( x \), square it, and add it to both sides.** 4. **Rewrite the left side as a perfect square trinomial.** 5. **Take the square root of both sides.** 6. **Solve for \( x \).** This method not only helps in solving quadratic equations but also in understanding the properties of parabolas and the derivation of the quadratic formula.