Solve each equation. See Examples 4–6. √(2√(7x+2)) = √(3x+2)

Question

Accepted Answer

Hey everyone in this problem for the following equation were asked to solve for X. We're given the equation. The square root of three. Square root of eight X plus five is equal to the square root of four. X plus five. Let's just write what we were given. So we have the square root of three. And inside of that we also have the square root of eight X plus five. Okay? And then on the right hand side we have the square root of four. X plus five. Okay now we're off to solve for X. And this equation looks scary at first but let's just break it down piece by piece. Okay? So first things first we have this elder square root on both the left and right sides. Okay so let's go ahead square both sides. We're gonna have three Times the square root of eight x plus five Is equal to four x plus five. All right now on the left hand side we still have this square root. So let's go ahead and square both sides one more time. We're gonna get three times the square root eight X plus five. All squared is equal to four X plus five. All squared. Now the left hand side, we get this three. Remember that we can distribute the exponent to these two things separately because they're being multiplied. So we get three squared which is going to give us nine times the square root of eight X plus five. If we square that then we're going to get eight X plus five. Okay? The square root is going to cancel with that square on the right hand side we have four X plus five. All squared. Okay this is gonna give us 16 x squared plus 40 X Plus 25. We expand. Alright. And now you'll start to realize that this just looks like a quadratic function. Okay? We have an X squared and X. And a constant term on the right hand side. We have an X. Term and a constant on the left hand side. Okay? So we want to solve this like we would with any quadratic equation. Okay so we're gonna expand on the left and then we're gonna move everything over to the one side. So nine times eight X. Is gonna give us 72 X. And nine times five is going to give us 45 on the right hand side, we still have 16 X squared plus 40 X plus 25. Moving everything over to the right hand side, we get 16 X squared -32 x - and that's all equal to zero. Alrighty. So now in order to solve for X, we know that we want to find um the roots. Okay? Or we wanna factor if we can factor then we can find the values of X. That solve this easily. So what we'll notice is that each of these terms can be divided by four. Okay so let's go ahead and do that. If we divide the left hand side by four, we're still going to get zero and when we divide the right hand side by four we get four X squared -8 X -5. Okay, that just gives us smaller numbers to work with when we're factoring, which can make it a little bit simpler. All right, so if we go ahead and factor this, you can factor this any way you'd like. Okay, and again if you have a hard time if you're not being able to factor it, if you're not seeing the factors or factoring is not working out, you can always use your quadratic formula to find the values of X to find those roots. Okay? So if you're stuck you can always use that. Alright, so the factors are gonna be two X -5. Okay. And two X plus one. Now this is all equal to zero. Okay, so if the first factor is equal to zero then the system is satisfied or this equation satisfied because we'll have zero equals zero and if the second factor is equal to zero we're gonna have the same thing. Okay, so our x values are going to come from when each of these is zero and so we're going to get X is equal to five halves from this first term and X is equal to negative a half from the second term Setting those individually equal to zero and those are gonna be our solutions. Okay, so we go back up five halves and negative a half. Now, the only thing you want to check when you're doing a problem like this. When you have square roots remember that? We cannot take the square root of a negative number. Okay. The square root of a negative number is not going to be real. So we just have to double check that if we substitute in our choices, our answer choices that we don't have a negative here. Okay. So our answers that we found five paths and negative half. So let me just write them up here. So we don't forget five halves and negative a half. Ok. And we just want to double check. So if we put five halves into either of these it's a positive value. So we're gonna get a positive value back because we're adding and multiplying. Okay for negative a half. If we put negative a half into four X plus five, four times negative a half it's going to be negative two. Then we add five. It's gonna be three so it's positive and same thing for eight X plus five. Okay, negative half times eight Is going to give us -4. We add five it's going to be positive. So those solutions are okay. All right. So if we're looking at our answer choices, we found that the solutions are going to be X equals negative half and X equals five halves and that's going to be answered. A thanks everyone for watching. I hope this video helped you in the next one.

Solve each equation. See Examples 4–6. √(2√(7x+2)) = √(3x+2)

Key Concepts

Square Roots

Isolating Variables

Equating Expressions

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