Solve for x in the equation x squared + 11 x + StartFraction 121 Over 4 EndFraction = StartFraction 125 Over 4 EndFraction.
x = negative 11 plus-or-minus StartFraction 25 Over 2 EndFraction
x = negative eleven-halves plus-or-minus StartFraction 25 Over 2 EndFraction
x = negative 11 plus-or-minus StartFraction 5 StartRoot 5 EndRoot Over 2 EndFraction
x = negative eleven-halves plus-or-minus StartFraction 5 StartRoot 5 EndRoot Over 2 EndFraction

We are given the quadratic equation:

\[
x^2 + 11x + \frac{121}{4} = \frac{125}{4}
\]

### Step 1: Simplify the equation

First, subtract \(\frac{121}{4}\) from both sides:

\[
x^2 + 11x = \frac{125}{4} - \frac{121}{4}
\]

\[
x^2 + 11x = \frac{4}{4} = 1
\]

Now, the equation is:

\[
x^2 + 11x = 1
\]

### Step 2: Complete the square

We want to complete the square for the left-hand side. Start by taking half of the coefficient of \(x\) (which is 11), then squaring it:

\[
\left(\frac{11}{2}\right)^2 = \frac{121}{4}
\]

Add \(\frac{121}{4}\) to both sides:

\[
x^2 + 11x + \frac{121}{4} = 1 + \frac{121}{4}
\]

This simplifies to:

\[
x^2 + 11x + \frac{121}{4} = \frac{125}{4}
\]

Now we have:

\[
\left(x + \frac{11}{2}\right)^2 = \frac{125}{4}
\]

### Step 3: Solve for \(x\)

Take the square root of both sides:

\[
x + \frac{11}{2} = \pm \frac{\sqrt{125}}{2}
\]

Since \(\sqrt{125} = 5\sqrt{5}\), this becomes:

\[
x + \frac{11}{2} = \pm \frac{5\sqrt{5}}{2}
\]

### Step 4: Isolate \(x\)

Now, solve for \(x\) by subtracting \(\frac{11}{2}\) from both sides:

\[
x = -\frac{11}{2} \pm \frac{5\sqrt{5}}{2}
\]

Thus, the solution is:

\[
x = -\frac{11}{2} \pm \frac{5\sqrt{5}}{2}
\]

### Conclusion:

The correct answer is:
\[
x = -\frac{11}{2} \pm \frac{5\sqrt{5}}{2}
\]

Would you like any further details or have any questions?

Here are five related questions:
1. How do you complete the square for a general quadratic equation?
2. What is the discriminant of a quadratic equation, and how is it used?
3. How do you solve a quadratic equation using the quadratic formula?
4. What are some methods to approximate square roots, like \(\sqrt{125}\)?
5. How do you graph quadratic functions, and what does the graph tell you about the solutions?

**Tip**: When completing the square, always ensure you balance both sides by adding or subtracting the same value to maintain equality.

Solve for x in the equation x^2 + 11x + (121/4) = (125/4). x = -11 ± (25/2), x = -11/2 ± (25/2), x = -11 ± (5√5/2), x = -11/2 ± (5√5/2).

Learn how to solve the quadratic equation x^2 + 11x + (121/4) = (125/4) using the completing the square method. This detailed step-by-step guide covers key algebraic concepts like completing the square and the quadratic formula. Perfect for students in Grades 9-10.

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