Solve the following quadratic equations by completing the square: 1. x² - 5x + 6 = 0; 2. x² + 2x = 8.

Steps to Solve

1. Solve the first equation: $x^2 + 5x + 6 = 0$

To complete the square, first rearrange the equation: $$ x^2 + 5x = -6 $$

Next, find the term to complete the square. Take half of the coefficient of $x$, which is $\frac{5}{2}$, and square it: $$ \left(\frac{5}{2}\right)^2 = \frac{25}{4} $$

Add $\frac{25}{4}$ to both sides: $$ x^2 + 5x + \frac{25}{4} = -6 + \frac{25}{4} $$

Convert $-6$ into quarters: $$ -6 = -\frac{24}{4} $$

Combine the right side: $$ x^2 + 5x + \frac{25}{4} = \frac{1}{4} $$

Now rewrite the left side as a square: $$ \left(x + \frac{5}{2}\right)^2 = \frac{1}{4} $$

Take the square root of both sides: $$ x + \frac{5}{2} = \pm \frac{1}{2} $$

Solve for $x$: $$ x = -\frac{5}{2} + \frac{1}{2} \quad \text{or} \quad x = -\frac{5}{2} - \frac{1}{2} $$

This gives: $$ x = -2 \quad \text{or} \quad x = -3 $$

2. Solve the second equation: $x^2 + 2x = 8$

Rearrange the equation: $$ x^2 + 2x - 8 = 0 $$

Now isolate $x^2 + 2x$: $$ x^2 + 2x = 8 $$

Next, find the term to complete the square. Take half of the coefficient of $x$, which is $1$, and square it: $$ 1^2 = 1 $$

Add $1$ to both sides: $$ x^2 + 2x + 1 = 8 + 1 $$

This simplifies to: $$ (x + 1)^2 = 9 $$

Take the square root of both sides: $$ x + 1 = \pm 3 $$

Solve for $x$: $$ x = -1 + 3 \quad \text{or} \quad x = -1 - 3 $$

This gives: $$ x = 2 \quad \text{or} \quad x = -4 $$