Question

5. Divide and simplify if possible.

√250x16√2x

6. What is the solution of the equation?

√2x+13−5=x

Answer 1

5. (5x^7)√(5x)

6. x = -2

Explanation:

5.

`\displaystyle\frac{\sqrt{250x^(16)}}{√(2x)}=\sqrt{(250x^(16))/(2x)}=\sqrt{125x^(15)}\\\\=\sqrt{(25x^(14))(5x)}=5x^7√(5x)`

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6.

`√(2x+13)-5=x\\\\√(2x+13)=x+5 \qquad\text{add 5}\\\\2x+13=x^2+10x+25 \qquad\text{square both sides}\\\\x^2+8x+12=0 \qquad\text{subtract the left side}\\\\(x+6)(x+2)=0 \qquad\text{factor}\\\\x=-6 \quad\text{or}\quad x=-2`

There is always the possibility of extraneous solutions for equations like this. We can check.

√(2·(-6)+13) -5 = -6 . . . . substitute -6 for x

√1 -5 = -6 . . . . . . . . . . . . not true; -6 is an extraneous solution

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√2·(-2)+13) -5 = -2

√9 -5 = -2 . . . . . . . . . . . true; x = -2 is the solution

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The attached graph has the equation rewritten so it is of the form

f(x) = 0

where

f(x) = √(2x+13) -5 -x

The x-intercept is highlighted on the graph. It is x=-2. You can see that if the negative branch of the square root function were included, it might make an x-intercept at x=-6. For our purposes, the square root function is the positive square root only, so that branch is not included and -6 is an extraneous solution.