Let's solve the quadratic equation [tex]\[x^2 + 11x + \frac{121}{4} = \frac{125}{4}.\][/tex]
1. Subtract [tex]\(\frac{125}{4}\)[/tex] from both sides to form a standard quadratic equation:
[tex]\[
x^2 + 11x + \frac{121}{4} - \frac{125}{4} = 0
\][/tex]
2. Combine like terms on the left side:
[tex]\[
x^2 + 11x + \frac{121 - 125}{4} = 0
\][/tex]
[tex]\[
x^2 + 11x - 1 = 0
\][/tex]
3. Solve the quadratic equation [tex]\(x^2 + 11x - 1 = 0\)[/tex].
We use the quadratic formula [tex]\(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 11\)[/tex], and [tex]\(c = -1\)[/tex].
First, calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = 11^2 - 4(1)(-1) = 121 + 4 = 125
\][/tex]
Next, find the solutions using the quadratic formula:
[tex]\[
x = \frac{{-11 \pm \sqrt{125}}}{2}
\][/tex]
Notice that [tex]\(\sqrt{125} = 5\sqrt{5}\)[/tex]. So the solutions can be rewritten as:
[tex]\[
x = \frac{{-11 \pm 5\sqrt{5}}}{2}
\][/tex]
Thus, there are two solutions:
[tex]\[
x_1 = \frac{{-11 + 5\sqrt{5}}}{2} \quad \text{and} \quad x_2 = \frac{{-11 - 5\sqrt{5}}}{2}
\][/tex]
4. Evaluate the given options:
- [tex]\(x = -11 \pm \frac{25}{2}\)[/tex]:
- [tex]\(-11 + \frac{25}{2} = -11 + 12.5 = 1.5\)[/tex]
- [tex]\(-11 - \frac{25}{2} = -11 - 12.5 = -23.5\)[/tex]
- [tex]\(x = -\frac{11}{2} \pm \frac{25}{2}\)[/tex]:
- This would correspond to:
- [tex]\(\frac{-11 + 25}{2}\)[/tex]
- [tex]\(\frac{-11 - 25}{2}\)[/tex]
Which would be incorrect, as the discriminant terms differ.
- [tex]\(x = -11 \pm \frac{5\sqrt{5}}{2}\)[/tex]:
- [tex]\(-11 + \frac{5\sqrt{5}}{2}\)[/tex]
- [tex]\(-11 - \frac{5\sqrt{5}}{2}\)[/tex]
- [tex]\(x = -\frac{11}{2} \pm \frac{5\sqrt{5}}{2}\)[/tex]:
- These represent:
- [tex]\(\frac{-11 + 5\sqrt{5}}{2}\)[/tex]
- [tex]\(\frac{-11 - 5\sqrt{5}}{2}\)[/tex]
But we need our solutions [tex]\(x_1 = \frac{-11 + 5\sqrt{5}}{2}\)[/tex] and [tex]\(x_2 = \frac{-11 - 5\sqrt{5}}{2}\)[/tex].
Final Correct Options:
[tex]\[
x = -11 \pm \frac{5\sqrt{5}}{2} \quad \text{and} \quad x = -\frac{11}{2} \pm \frac{5\sqrt{5}}{2}
\][/tex]
---
Our numerical results are approximately [tex]\(x \approx -11.0901699437495\)[/tex] and [tex]\(x \approx 0.0901699437494742\)[/tex] after evaluating these expressions directly. So [tex]\(x\)[/tex] values must align with [tex]\(-\frac{11 + 5\sqrt{5}}{2}\)[/tex] approximately [tex]\(-5.5+\pm 5\sqrt{5}=1.58358\pm-23.5\)[/tex], matching correct bracketed results:
[tex]\[ (-5.5+\pm 5)/\sqrt(5)+4,5*\sqrt(4,5-(-1.4/2)=(-\approx \1.5,\approx -23.5)\][/tex]+=[-, .
