Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
Sagot :
To determine which equation can be rewritten as [tex]\( x + 4 = x^2 \)[/tex] with the condition [tex]\( x > 0 \)[/tex], we will analyze each given equation step by step.
### Equation 1: [tex]\(\sqrt{x} + 2 = x\)[/tex]
1. Start with [tex]\(\sqrt{x} + 2 = x\)[/tex].
2. Isolate the square root term: [tex]\(\sqrt{x} = x - 2\)[/tex].
3. Square both sides: [tex]\(x = (x - 2)^2\)[/tex].
4. Expand the right side: [tex]\(x = x^2 - 4x + 4\)[/tex].
5. Rearrange the equation: [tex]\(x = x^2 - 4x + 4\)[/tex].
6. Collect all terms to one side: [tex]\(x^2 - 5x + 4 = 0\)[/tex].
7. Factor the quadratic: [tex]\((x - 1)(x - 4) = 0\)[/tex].
8. Solve for [tex]\(x\)[/tex]: [tex]\(x = 1\)[/tex] or [tex]\(x = 4\)[/tex].
With [tex]\( x > 0 \)[/tex], both solutions [tex]\( x = 1 \)[/tex] and [tex]\( x = 4 \)[/tex] are valid solutions, but we will determine if the equation fits [tex]\( x + 4 = x^2 \)[/tex] later.
### Equation 2: [tex]\(\sqrt{x + 2} = x\)[/tex]
1. Start with [tex]\(\sqrt{x + 2} = x\)[/tex].
2. Square both sides: [tex]\(x + 2 = x^2\)[/tex].
3. Rearrange the equation: [tex]\(x^2 - x - 2 = 0\)[/tex].
4. Factor the quadratic: [tex]\((x - 2)(x + 1) = 0\)[/tex].
5. Solve for [tex]\(x\)[/tex]: [tex]\(x = 2\)[/tex] or [tex]\(x = -1\)[/tex].
With [tex]\( x > 0 \)[/tex], the valid solution is [tex]\( x = 2 \)[/tex].
Check how it fits with [tex]\(x + 4 = x^2\)[/tex]:
- For [tex]\( x = 2 \)[/tex]: [tex]\( 2 + 4 = 6 \)[/tex] and [tex]\( 2^2 = 4 \)[/tex]. It does not fit.
### Equation 3: [tex]\(\sqrt{x + 4} = x\)[/tex]
1. Start with [tex]\(\sqrt{x + 4} = x\)[/tex].
2. Square both sides: [tex]\(x + 4 = x^2\)[/tex].
3. Rearrange: [tex]\(x^2 - x - 4 = 0\)[/tex].
This is the form [tex]\(x + 4 = x^2\)[/tex]!
Solve it to check potential [tex]\(x\)[/tex]:
- Factorization might be tedious so quadratic solution:
[tex]\[x = \frac{1 \pm \sqrt{1 + 16}}{2} = \frac{1 \pm \sqrt{17}}{2}\][/tex]
With [tex]\( x > 0 \)[/tex], only positive [tex]\(x = \frac{1 + \sqrt{17}}{2}\)[/tex].
### Equation 4: [tex]\(\sqrt{x^2 + 16} = x\)[/tex]
1. Start with [tex]\(\sqrt{x^2 + 16} = x\)[/tex].
2. Square both sides: [tex]\(x^2 + 16 = x^2\)[/tex].
3. Simplify: [tex]\(16 = 0\)[/tex].
This provides no valid [tex]\(x\)[/tex].
### Conclusion
From the analysis, the equation that can be rewritten as [tex]\( x + 4 = x^2 \)[/tex] is:
[tex]\[ \sqrt{x + 4} = x \][/tex]
### Equation 1: [tex]\(\sqrt{x} + 2 = x\)[/tex]
1. Start with [tex]\(\sqrt{x} + 2 = x\)[/tex].
2. Isolate the square root term: [tex]\(\sqrt{x} = x - 2\)[/tex].
3. Square both sides: [tex]\(x = (x - 2)^2\)[/tex].
4. Expand the right side: [tex]\(x = x^2 - 4x + 4\)[/tex].
5. Rearrange the equation: [tex]\(x = x^2 - 4x + 4\)[/tex].
6. Collect all terms to one side: [tex]\(x^2 - 5x + 4 = 0\)[/tex].
7. Factor the quadratic: [tex]\((x - 1)(x - 4) = 0\)[/tex].
8. Solve for [tex]\(x\)[/tex]: [tex]\(x = 1\)[/tex] or [tex]\(x = 4\)[/tex].
With [tex]\( x > 0 \)[/tex], both solutions [tex]\( x = 1 \)[/tex] and [tex]\( x = 4 \)[/tex] are valid solutions, but we will determine if the equation fits [tex]\( x + 4 = x^2 \)[/tex] later.
### Equation 2: [tex]\(\sqrt{x + 2} = x\)[/tex]
1. Start with [tex]\(\sqrt{x + 2} = x\)[/tex].
2. Square both sides: [tex]\(x + 2 = x^2\)[/tex].
3. Rearrange the equation: [tex]\(x^2 - x - 2 = 0\)[/tex].
4. Factor the quadratic: [tex]\((x - 2)(x + 1) = 0\)[/tex].
5. Solve for [tex]\(x\)[/tex]: [tex]\(x = 2\)[/tex] or [tex]\(x = -1\)[/tex].
With [tex]\( x > 0 \)[/tex], the valid solution is [tex]\( x = 2 \)[/tex].
Check how it fits with [tex]\(x + 4 = x^2\)[/tex]:
- For [tex]\( x = 2 \)[/tex]: [tex]\( 2 + 4 = 6 \)[/tex] and [tex]\( 2^2 = 4 \)[/tex]. It does not fit.
### Equation 3: [tex]\(\sqrt{x + 4} = x\)[/tex]
1. Start with [tex]\(\sqrt{x + 4} = x\)[/tex].
2. Square both sides: [tex]\(x + 4 = x^2\)[/tex].
3. Rearrange: [tex]\(x^2 - x - 4 = 0\)[/tex].
This is the form [tex]\(x + 4 = x^2\)[/tex]!
Solve it to check potential [tex]\(x\)[/tex]:
- Factorization might be tedious so quadratic solution:
[tex]\[x = \frac{1 \pm \sqrt{1 + 16}}{2} = \frac{1 \pm \sqrt{17}}{2}\][/tex]
With [tex]\( x > 0 \)[/tex], only positive [tex]\(x = \frac{1 + \sqrt{17}}{2}\)[/tex].
### Equation 4: [tex]\(\sqrt{x^2 + 16} = x\)[/tex]
1. Start with [tex]\(\sqrt{x^2 + 16} = x\)[/tex].
2. Square both sides: [tex]\(x^2 + 16 = x^2\)[/tex].
3. Simplify: [tex]\(16 = 0\)[/tex].
This provides no valid [tex]\(x\)[/tex].
### Conclusion
From the analysis, the equation that can be rewritten as [tex]\( x + 4 = x^2 \)[/tex] is:
[tex]\[ \sqrt{x + 4} = x \][/tex]
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Stay curious and keep coming back to Westonci.ca for answers to all your burning questions.
