Complete the square \( 2 x^{2}+7 x+6 \)

To complete the square for the quadratic expression \( 2x^{2} + 7x + 6 \), follow these steps: 1. **Factor Out the Leading Coefficient:** Start by factoring out the coefficient of \( x^2 \) from the first two terms: \[ 2x^{2} + 7x + 6 = 2\left(x^{2} + \frac{7}{2}x\right) + 6 \] 2. **Complete the Square Inside the Parentheses:** - Take half of the coefficient of \( x \), which is \( \frac{7}{2} \times \frac{1}{2} = \frac{7}{4} \). - Square this value: \( \left(\frac{7}{4}\right)^2 = \frac{49}{16} \). Add and subtract this squared term inside the parentheses to complete the square: \[ 2\left(x^{2} + \frac{7}{2}x + \frac{49}{16} - \frac{49}{16}\right) + 6 \] \[ = 2\left(\left(x + \frac{7}{4}\right)^2 - \frac{49}{16}\right) + 6 \] 3. **Simplify the Expression:** Distribute the 2 and combine like terms: \[ = 2\left(x + \frac{7}{4}\right)^2 - 2 \times \frac{49}{16} + 6 \] \[ = 2\left(x + \frac{7}{4}\right)^2 - \frac{49}{8} + \frac{48}{8} \] \[ = 2\left(x + \frac{7}{4}\right)^2 - \frac{1}{8} \] 4. **Final Answer:** \[ 2x^{2} + 7x + 6 = 2\left(x + \frac{7}{4}\right)^2 - \frac{1}{8} \]