At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Get quick and reliable solutions to your questions from a community of seasoned experts on our user-friendly platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To determine which of the provided options are solutions to the equation [tex]\(x^2 - 10x + 25 = 17\)[/tex], we'll follow these steps:
1. Simplify the equation:
[tex]\[ x^2 - 10x + 25 = 17 \][/tex]
Subtract 17 from both sides to set the equation to zero:
[tex]\[ x^2 - 10x + 25 - 17 = 0 \][/tex]
This simplifies to:
[tex]\[ x^2 - 10x + 8 = 0 \][/tex]
2. Factor or use the quadratic formula:
The quadratic formula for [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the equation [tex]\(x^2 - 10x + 8 = 0\)[/tex]:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -10\)[/tex]
- [tex]\(c = 8\)[/tex]
Plug these values into the quadratic formula:
[tex]\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} \][/tex]
Simplify the expression under the square root:
[tex]\[ x = \frac{10 \pm \sqrt{100 - 32}}{2} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{68}}{2} \][/tex]
[tex]\[ x = \frac{10 \pm 2\sqrt{17}}{2} \][/tex]
Simplify the fractions:
[tex]\[ x = 5 \pm \sqrt{17} \][/tex]
So, the solutions to the equation [tex]\(x^2 - 10x + 8 = 0\)[/tex] are:
[tex]\[ x = 5 + \sqrt{17} \quad \text{and} \quad x = 5 - \sqrt{17} \][/tex]
3. Check the provided options:
- A. [tex]\(x = \sqrt{17} - 5\)[/tex]: This is not one of our solutions.
- B. [tex]\(x = \sqrt{8} + 5\)[/tex]: This is not one of our solutions.
- C. [tex]\(x = -\sqrt{8} - 5\)[/tex]: This is not one of our solutions.
- D. [tex]\(x = \sqrt{17} + 5\)[/tex]: This matches [tex]\(5 + \sqrt{17}\)[/tex].
- E. [tex]\(x = -\sqrt{17} - 5\)[/tex]: This is not one of our solutions.
- F. [tex]\(x = -\sqrt{17} + 5\)[/tex]: This matches [tex]\(5 - \sqrt{17}\)[/tex].
Based on this analysis, the solutions to the equation [tex]\(x^2 - 10x + 25 = 17\)[/tex] are:
- D. [tex]\(x = \sqrt{17} + 5\)[/tex]
- F. [tex]\(x = -\sqrt{17} + 5\)[/tex]
1. Simplify the equation:
[tex]\[ x^2 - 10x + 25 = 17 \][/tex]
Subtract 17 from both sides to set the equation to zero:
[tex]\[ x^2 - 10x + 25 - 17 = 0 \][/tex]
This simplifies to:
[tex]\[ x^2 - 10x + 8 = 0 \][/tex]
2. Factor or use the quadratic formula:
The quadratic formula for [tex]\(ax^2 + bx + c = 0\)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the equation [tex]\(x^2 - 10x + 8 = 0\)[/tex]:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -10\)[/tex]
- [tex]\(c = 8\)[/tex]
Plug these values into the quadratic formula:
[tex]\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 8}}{2 \cdot 1} \][/tex]
Simplify the expression under the square root:
[tex]\[ x = \frac{10 \pm \sqrt{100 - 32}}{2} \][/tex]
[tex]\[ x = \frac{10 \pm \sqrt{68}}{2} \][/tex]
[tex]\[ x = \frac{10 \pm 2\sqrt{17}}{2} \][/tex]
Simplify the fractions:
[tex]\[ x = 5 \pm \sqrt{17} \][/tex]
So, the solutions to the equation [tex]\(x^2 - 10x + 8 = 0\)[/tex] are:
[tex]\[ x = 5 + \sqrt{17} \quad \text{and} \quad x = 5 - \sqrt{17} \][/tex]
3. Check the provided options:
- A. [tex]\(x = \sqrt{17} - 5\)[/tex]: This is not one of our solutions.
- B. [tex]\(x = \sqrt{8} + 5\)[/tex]: This is not one of our solutions.
- C. [tex]\(x = -\sqrt{8} - 5\)[/tex]: This is not one of our solutions.
- D. [tex]\(x = \sqrt{17} + 5\)[/tex]: This matches [tex]\(5 + \sqrt{17}\)[/tex].
- E. [tex]\(x = -\sqrt{17} - 5\)[/tex]: This is not one of our solutions.
- F. [tex]\(x = -\sqrt{17} + 5\)[/tex]: This matches [tex]\(5 - \sqrt{17}\)[/tex].
Based on this analysis, the solutions to the equation [tex]\(x^2 - 10x + 25 = 17\)[/tex] are:
- D. [tex]\(x = \sqrt{17} + 5\)[/tex]
- F. [tex]\(x = -\sqrt{17} + 5\)[/tex]
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Westonci.ca is your go-to source for reliable answers. Return soon for more expert insights.