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lukranit [14]
3 years ago
11

What is the solution of x=2+ square root x-2?

Mathematics
1 answer:
alex41 [277]3 years ago
8 0

So I'm going to assume that this question is asking for <u>non extraneous solutions</u>, or solutions that are found in the equation <em>and</em> are valid solutions when plugged back into the equation. So firstly, subtract 2 on both sides of the equation:

x-2=\sqrt{x-2}

Next, square both sides:

(x-2)^2=x-2\\(x-2)(x-2)=x-2\\x^2-4x+4=x-2

Next, subtract x and add 2 to both sides of the equation:

x^2-5x+6=0

Now we are going to be factoring by grouping to find the solution(s). Firstly, what two terms have a product of 6x^2 and a sum of -5x? That would be -3x and -2x. Replace -5x with -2x - 3x:

x^2-2x-3x+6=0

Next, factor x^2 - 2x and -3x + 6 separately. Make sure that they have the same quantity on the inside of the parentheses:

x(x-2)-3(x-2)=0

Now you can rewrite the equation as (x-3)(x-2)=0

Now, apply the Zero Product Property and solve for x as such:

x-3=0\\x=3\\\\x-2=0\\x=2

Now, it may appear that the answer is C, however we need to plug the numbers back into the original equation to see if they are true as such:

2=2+\sqrt{2-2}\\2=2+\sqrt{0}\\2=2+0\\2=2\ \textsf{true}\\\\3=2+\sqrt{3-2}\\3=2+\sqrt{1}\\3=2+1\\3=3\ \textsf{true}

Since both solutions hold true when x = 2 and x = 3, <u>your answer is C. x = 2 or x = 3.</u>

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