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In $3-38,$ solve each equation for the variable, check, and write the solution set.$$\sqrt{5 x}-\sqrt{x+4}=3$$

$\left\{\frac{35+15 \sqrt{5}}{8}, \frac{35-15 \sqrt{5}}{8}\right\}$

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Chapter 3

REAL NUMBERS AND RADICALS

Section 8

Solving Radical Equations

Whole which of Numbers

Fractions and Mixed Numbers

Decimals

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In $3-38,$ solve each equa…

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If you have the equation here, X is equal to the square root of four X plus five. So to go ahead and to basically eliminate the square root, go ahead and square both sides of the equation. So we just get well, X square is X squared, and that's equal to the square root of four experts, I swear, which is just equal to what's under the radical equal to four X plus five. Okay, so then we can go ahead and against you make this a quadratic in the standard form by sending this equal to zero. So subtract all our terms to the left side. We get that X squared minus four X and then minus five is equal to zero. Okay, so we have a quadratic equal to zero. We can try and go ahead and factor this and I believe this is gonna factor So we can't this factors as well. X squared x Times X, not five. That's five times one right. And then, well, we need a minus four x in the middle of self. A bit of minus on the five plus on the one that's an X squared plus one x minus five X minus four X and then minus five times one minus five. Okay, that's factored correctly. And that is that equal to zero. So we see here that X would be equal to either five or X would be equal to negative one. Okay, but let's go ahead. And it's actually check these solutions to make sure they actually work in our original equation here. So exited five. Well, then we have five is equal to the square root of four times five plus five. So we have five is equal to the square root of what times five is 20 plus five is 25. So we have five is equal to the square root of 25 which is true, right? Because the sport 25 is five. Therefore, we have five is equal to five. Check now, what about if access is a negative one? But then we have negative one is equal to four times negative. One plus five. Okay, so we'd have a negative one. Negative one is equal to the square root of of four times negative. One plus 54 times negative. One plus five. Okay. Would have negative one is equal to the square. Root off four times. Negative one is negative. Four plus five. Okay, we have negative. One is equal to while the square root of negative 4.5 is positive one. So we have one could have negative one is equal to the square. Root off one. Um well, right. The square root of one is one would have native one is equal toe one and do that is not true. Right? So therefore, our solutions that here is we have a vision, extraneous solution. Our solution sets would be just to set containing five so right that as a set containing the number first, right?

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