Solve for [tex]\( x \)[/tex] in the equation [tex]\( x^2 + 11x + \frac{121}{4} = \frac{125}{4} \)[/tex].

A. [tex]\( x = -11 \pm \frac{25}{2} \)[/tex]

B. [tex]\( x = -\frac{11}{2} \pm \frac{25}{2} \)[/tex]

C. [tex]\( x = -11 \pm \frac{5 \sqrt{5}}{2} \)[/tex]

D. [tex]\( x = -\frac{11}{2} \pm \frac{5 \sqrt{5}}{2} \)[/tex]



Answer :

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]

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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]+=[-, .