Question

Complete the square to solve the quadratic equation.
xsquaredminus11xplus30equals0
Question content area bottom
Part 1
xequals
  
enter your response here
​(Simplify your answer. Type an exact​ answer, using radicals as needed. Use a comma to separate answers as​ needed.)

Answer 1

To solve the quadratic equation \( x^2 - 11x + 30 = 0 \) by completing the square, follow these steps:

1. **Move the constant to the other side:**
\[
x^2 - 11x = -30
\]

2. **Complete the square:**
- Take half of the coefficient of \( x \) (which is -11), square it, and add it to both sides. Half of -11 is -\(\frac{11}{2}\), and squaring it gives:
\[
\left(-\frac{11}{2}\right)^2 = \frac{121}{4}
\]
- Add \( \frac{121}{4} \) to both sides:
\[
x^2 - 11x + \frac{121}{4} = -30 + \frac{121}{4}
\]

3. **Simplify the right side:**
- Convert -30 to a fraction with a denominator of 4:
\[
-30 = -\frac{120}{4}
\]
- Now add:
\[
-30 + \frac{121}{4} = \frac{-120 + 121}{4} = \frac{1}{4}
\]

4. **Rewrite the left side as a square:**
\[
\left(x - \frac{11}{2}\right)^2 = \frac{1}{4}
\]

5. **Take the square root of both sides:**
\[
x - \frac{11}{2} = \pm \frac{1}{2}
\]

6. **Solve for \( x \):**
- For the positive case:
\[
x - \frac{11}{2} = \frac{1}{2} \implies x = \frac{1}{2} + \frac{11}{2} = \frac{12}{2} = 6
\]
- For the negative case:
\[
x - \frac{11}{2} = -\frac{1}{2} \implies x = -\frac{1}{2} + \frac{11}{2} = \frac{10}{2} = 5
\]

So, the solutions to the equation \( x^2 - 11x + 30 = 0 \) are:
\[
x = 6, 5
\]
Finally, write your answer as:
\[
x = 5, 6
\]