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Problem 29

Central Michigan University

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Problem 25

$17-28=$ Completing the Square Find all real solutions of the

equation by completing the square.

$$5 x^{2}+10 x-7=0$$

Answer

$x=-1 \pm \frac{2 \sqrt{15}}{5}$

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## Discussion

## Video Transcript

Okay. Completing the square. This gets a little bit tricky. They get tougher than this even. But try and walk you through this here stuff by step. I have already completed this problem, and I have checked my answer already. So do you know it's right. Here we go. So the first thing I gotta do is add seven double sides of this to solve by completing the square, Um, and I've got to give the X squared and the ex term alone in the left. So added seven to both sides. Now I'm going to divide everything by five because I want my lead coefficient to be one. Okay, so now that I've got the lead coefficient one, I can actually complete the square. So the process of completing the square is the idea of taking half the be value. You can only do this with legal officials. 1/2 of two is one. And then you square it once where this one So I'm gonna add one to both sides of this equation. There are adding one to both sides. The question The only reason I did 5 50 because I wanted a common denominator. Okay, So why did I have one? Because now I've got a perfect square trying no meal over here. And they left expert plus two x plus one, which could be factored easily as X plus one squared on the right. I'll just simplify. 7/5 um, plus 5/5 is 12 5th The reason I wanted affected that, like that is so I could turn it into X S O. I could take the square root of both sides. All right. The square root of, um, expose one squared is just explosive. One in the square root of 12 5th is plus or minus squared of 12 5th Then I can I subtract one from both sides and I get kind of my answer. All right, so, um, negative one plus or minus the square to 12 5th Usually we don't leave fractions. Insider radicals, though that's not considered simplified. There's an easy fix, though. If if you just multiplied by the square to five over the square to five, that'll be that'll rationalize that, Um, the denominator it's called and I'll give us this new expression negative one plus or minus the square root of 60 over five. Because on the bottom. You get squared five times, escorted five, which is five and scored 12 times scored. Five is the 12 squirt of 60. But I would even go further with this, and I would simplify that radical in the numerator sixties, divisible by perfect square, which is four four times fifteen's too square to 60 and square to forest too. So two times the square of 15 is equal to the square to 60 and ever five. So you have to ask your teacher what they're looking for out of you right now. But technically, this is the simplified answer. Negative one plus or minus two times the square of 15 all over five. They might just want you to go with this for now because I know you are in just chapter one of this book, so not sure how difficult don't want to make it. But technically, that's the simplified answer

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