C. by completing the square. 1. x² - 5x + 6 = 0 2. x² + 2x = 8

Steps to Solve

1. Rearranging the First Equation To start, we will rearrange the first equation to isolate the constant on one side: $$ x^2 - 5x + 6 = 0 $$ Subtract 6 from both sides: $$ x^2 - 5x = -6 $$

2. Completing the Square for the First Equation Next, to complete the square, take half of the coefficient of $x$ (which is $-5$), square it, and add it to both sides. Half of $-5$ is $-\frac{5}{2}$, and squaring it gives: $$ \left(-\frac{5}{2}\right)^2 = \frac{25}{4} $$ Now add $\frac{25}{4}$ to both sides: $$ x^2 - 5x + \frac{25}{4} = -6 + \frac{25}{4} $$

3. Simplifying the Right Side Convert $-6$ to a fraction: $$ -6 = -\frac{24}{4} $$ The equation simplifies to: $$ x^2 - 5x + \frac{25}{4} = \frac{1}{4} $$

4. Factoring the Left Side Now, factor the left side: $$ \left(x - \frac{5}{2}\right)^2 = \frac{1}{4} $$

5. Taking the Square Root Next, take the square root of both sides: $$ x - \frac{5}{2} = \pm \frac{1}{2} $$

6. Solving for $x$ Add $\frac{5}{2}$ to both sides:

• For the positive case: $$ x = \frac{5}{2} + \frac{1}{2} = 3 $$
• For the negative case: $$ x = \frac{5}{2} - \frac{1}{2} = 2 $$

7. Rearranging the Second Equation Now let's move to the second equation: $$ x^2 + 2x = 8 $$ Rearranging gives: $$ x^2 + 2x - 8 = 0 $$

8. Completing the Square for the Second Equation Take half of $2$, square it, and add it to both sides: $$ \left(1\right)^2 = 1 $$ Adding gives: $$ x^2 + 2x + 1 = 8 + 1 $$

9. Simplifying the Left Side This simplifies to: $$ (x + 1)^2 = 9 $$

10. Taking the Square Root for Second Equation Taking the square root of both sides gives: $$ x + 1 = \pm 3 $$

11. Solving for $x$ in the Second Equation

• For the positive case: $$ x = 3 - 1 = 2 $$
• For the negative case: $$ x = -3 - 1 = -4 $$