Connect with knowledgeable individuals and find the best answers at IDNLearn.com. Discover the information you need quickly and easily with our reliable and thorough Q&A platform.
Sagot :
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]+=[-, .
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]+=[-, .
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Thank you for choosing IDNLearn.com. We’re here to provide reliable answers, so please visit us again for more solutions.
