Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
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]+=[-, .
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
