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To determine which values of [tex]\(x\)[/tex] satisfy the equation [tex]\(4x^2 + 4x + 1 = 9\)[/tex], we will substitute each given value into the equation and check whether the left-hand side equals 9.
Let's check each option one by one:
Option A: [tex]\(x = \sqrt{10} + \frac{1}{2}\)[/tex]
1. Substitute [tex]\(x = \sqrt{10} + \frac{1}{2}\)[/tex] into the equation:
[tex]\[ 4\left(\sqrt{10} + \frac{1}{2}\right)^2 + 4\left(\sqrt{10} + \frac{1}{2}\right) + 1 \][/tex]
2. Calculate [tex]\(4\left(\sqrt{10} + \frac{1}{2}\right)^2\)[/tex]:
[tex]\[ 4\left(10 + \sqrt{10} + \frac{1}{4}\right) = 4\left(10.25 + \sqrt{10}\right) \approx 40.25 + 4\sqrt{10} \][/tex]
3. Calculate [tex]\(4\left(\sqrt{10} + \frac{1}{2}\right)\)[/tex]:
[tex]\[ 4\sqrt{10} + 2 \][/tex]
4. Combine these:
[tex]\[ 40.25 + 4\sqrt{10} + 4\sqrt{10} + 2 + 1 = 43.25 + 8\sqrt{10} \neq 9 \][/tex]
Thus, Option A does not satisfy the equation.
Option B: [tex]\(x = -2\)[/tex]
1. Substitute [tex]\(x = -2\)[/tex] into the equation:
[tex]\[ 4(-2)^2 + 4(-2) + 1 \][/tex]
2. Calculate [tex]\(4(-2)^2\)[/tex]:
[tex]\[ 4 \cdot 4 = 16 \][/tex]
3. Calculate [tex]\(4(-2)\)[/tex]:
[tex]\[ -8 \][/tex]
4. Combine these:
[tex]\[ 16 - 8 + 1 = 9 \][/tex]
Thus, Option B satisfies the equation.
Option C: [tex]\(x = \frac{\sqrt{2}}{2}\)[/tex]
1. Substitute [tex]\(x = \frac{\sqrt{2}}{2}\)[/tex] into the equation:
[tex]\[ 4\left(\frac{\sqrt{2}}{2}\right)^2 + 4\left(\frac{\sqrt{2}}{2}\right) + 1 \][/tex]
2. Calculate [tex]\(4\left(\frac{\sqrt{2}}{2}\right)^2\)[/tex]:
[tex]\[ 4 \cdot \frac{2}{4} = 2 \][/tex]
3. Calculate [tex]\(4\left(\frac{\sqrt{2}}{2}\right)\)[/tex]:
[tex]\[ 2\sqrt{2} \][/tex]
4. Combine these:
[tex]\[ 2 + 2\sqrt{2} + 1 \neq 9 \][/tex]
Thus, Option C does not satisfy the equation.
Option D: [tex]\(x = -\sqrt{10} + \frac{1}{2}\)[/tex]
1. Substitute [tex]\(x = -\sqrt{10} + \frac{1}{2}\)[/tex] into the equation:
[tex]\[ 4\left(-\sqrt{10} + \frac{1}{2}\right)^2 + 4\left(-\sqrt{10} + \frac{1}{2}\right) + 1 \][/tex]
2. Calculate [tex]\(4\left(-\sqrt{10} + \frac{1}{2}\right)^2\)[/tex]:
[tex]\[ 4\left(10 - \sqrt{10} + \frac{1}{4}\right) = 4\left(10.25 - \sqrt{10}\right) \approx 40.25 - 4\sqrt{10} \][/tex]
3. Calculate [tex]\(4\left(-\sqrt{10} + \frac{1}{2}\right)\)[/tex]:
[tex]\[ -4\sqrt{10} + 2 \][/tex]
4. Combine these:
[tex]\[ 40.25 - 4\sqrt{10} - 4\sqrt{10} + 2 + 1 = 43.25 - 8\sqrt{10} \neq 9 \][/tex]
Thus, Option D does not satisfy the equation.
Option E: [tex]\(x = 1\)[/tex]
1. Substitute [tex]\(x = 1\)[/tex] into the equation:
[tex]\[ 4(1)^2 + 4(1) + 1 \][/tex]
2. Calculate [tex]\(4(1)^2\)[/tex]:
[tex]\[ 4 \][/tex]
3. Calculate [tex]\(4(1)\)[/tex]:
[tex]\[ 4 \][/tex]
4. Combine these:
[tex]\[ 4 + 4 + 1 = 9 \][/tex]
Thus, Option E satisfies the equation.
Option F: [tex]\(x = \sqrt{2}\)[/tex]
1. Substitute [tex]\(x = \sqrt{2}\)[/tex] into the equation:
[tex]\[ 4(\sqrt{2})^2 + 4(\sqrt{2}) + 1 \][/tex]
2. Calculate [tex]\(4(\sqrt{2})^2\)[/tex]:
[tex]\[ 4 \cdot 2 = 8 \][/tex]
3. Calculate [tex]\(4(\sqrt{2})\)[/tex]:
[tex]\[ 4\sqrt{2} \][/tex]
4. Combine these:
[tex]\[ 8 + 4\sqrt{2} + 1 \neq 9 \][/tex]
Thus, Option F does not satisfy the equation.
### Conclusion
The solutions to the equation [tex]\(4x^2 + 4x + 1 = 9\)[/tex] are:
- [tex]\(x = -2\)[/tex]
- [tex]\(x = 1\)[/tex]
So the correct answers are:
B. [tex]\(x = -2\)[/tex]
E. [tex]\(x = 1\)[/tex]
Let's check each option one by one:
Option A: [tex]\(x = \sqrt{10} + \frac{1}{2}\)[/tex]
1. Substitute [tex]\(x = \sqrt{10} + \frac{1}{2}\)[/tex] into the equation:
[tex]\[ 4\left(\sqrt{10} + \frac{1}{2}\right)^2 + 4\left(\sqrt{10} + \frac{1}{2}\right) + 1 \][/tex]
2. Calculate [tex]\(4\left(\sqrt{10} + \frac{1}{2}\right)^2\)[/tex]:
[tex]\[ 4\left(10 + \sqrt{10} + \frac{1}{4}\right) = 4\left(10.25 + \sqrt{10}\right) \approx 40.25 + 4\sqrt{10} \][/tex]
3. Calculate [tex]\(4\left(\sqrt{10} + \frac{1}{2}\right)\)[/tex]:
[tex]\[ 4\sqrt{10} + 2 \][/tex]
4. Combine these:
[tex]\[ 40.25 + 4\sqrt{10} + 4\sqrt{10} + 2 + 1 = 43.25 + 8\sqrt{10} \neq 9 \][/tex]
Thus, Option A does not satisfy the equation.
Option B: [tex]\(x = -2\)[/tex]
1. Substitute [tex]\(x = -2\)[/tex] into the equation:
[tex]\[ 4(-2)^2 + 4(-2) + 1 \][/tex]
2. Calculate [tex]\(4(-2)^2\)[/tex]:
[tex]\[ 4 \cdot 4 = 16 \][/tex]
3. Calculate [tex]\(4(-2)\)[/tex]:
[tex]\[ -8 \][/tex]
4. Combine these:
[tex]\[ 16 - 8 + 1 = 9 \][/tex]
Thus, Option B satisfies the equation.
Option C: [tex]\(x = \frac{\sqrt{2}}{2}\)[/tex]
1. Substitute [tex]\(x = \frac{\sqrt{2}}{2}\)[/tex] into the equation:
[tex]\[ 4\left(\frac{\sqrt{2}}{2}\right)^2 + 4\left(\frac{\sqrt{2}}{2}\right) + 1 \][/tex]
2. Calculate [tex]\(4\left(\frac{\sqrt{2}}{2}\right)^2\)[/tex]:
[tex]\[ 4 \cdot \frac{2}{4} = 2 \][/tex]
3. Calculate [tex]\(4\left(\frac{\sqrt{2}}{2}\right)\)[/tex]:
[tex]\[ 2\sqrt{2} \][/tex]
4. Combine these:
[tex]\[ 2 + 2\sqrt{2} + 1 \neq 9 \][/tex]
Thus, Option C does not satisfy the equation.
Option D: [tex]\(x = -\sqrt{10} + \frac{1}{2}\)[/tex]
1. Substitute [tex]\(x = -\sqrt{10} + \frac{1}{2}\)[/tex] into the equation:
[tex]\[ 4\left(-\sqrt{10} + \frac{1}{2}\right)^2 + 4\left(-\sqrt{10} + \frac{1}{2}\right) + 1 \][/tex]
2. Calculate [tex]\(4\left(-\sqrt{10} + \frac{1}{2}\right)^2\)[/tex]:
[tex]\[ 4\left(10 - \sqrt{10} + \frac{1}{4}\right) = 4\left(10.25 - \sqrt{10}\right) \approx 40.25 - 4\sqrt{10} \][/tex]
3. Calculate [tex]\(4\left(-\sqrt{10} + \frac{1}{2}\right)\)[/tex]:
[tex]\[ -4\sqrt{10} + 2 \][/tex]
4. Combine these:
[tex]\[ 40.25 - 4\sqrt{10} - 4\sqrt{10} + 2 + 1 = 43.25 - 8\sqrt{10} \neq 9 \][/tex]
Thus, Option D does not satisfy the equation.
Option E: [tex]\(x = 1\)[/tex]
1. Substitute [tex]\(x = 1\)[/tex] into the equation:
[tex]\[ 4(1)^2 + 4(1) + 1 \][/tex]
2. Calculate [tex]\(4(1)^2\)[/tex]:
[tex]\[ 4 \][/tex]
3. Calculate [tex]\(4(1)\)[/tex]:
[tex]\[ 4 \][/tex]
4. Combine these:
[tex]\[ 4 + 4 + 1 = 9 \][/tex]
Thus, Option E satisfies the equation.
Option F: [tex]\(x = \sqrt{2}\)[/tex]
1. Substitute [tex]\(x = \sqrt{2}\)[/tex] into the equation:
[tex]\[ 4(\sqrt{2})^2 + 4(\sqrt{2}) + 1 \][/tex]
2. Calculate [tex]\(4(\sqrt{2})^2\)[/tex]:
[tex]\[ 4 \cdot 2 = 8 \][/tex]
3. Calculate [tex]\(4(\sqrt{2})\)[/tex]:
[tex]\[ 4\sqrt{2} \][/tex]
4. Combine these:
[tex]\[ 8 + 4\sqrt{2} + 1 \neq 9 \][/tex]
Thus, Option F does not satisfy the equation.
### Conclusion
The solutions to the equation [tex]\(4x^2 + 4x + 1 = 9\)[/tex] are:
- [tex]\(x = -2\)[/tex]
- [tex]\(x = 1\)[/tex]
So the correct answers are:
B. [tex]\(x = -2\)[/tex]
E. [tex]\(x = 1\)[/tex]
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