Solve: √(6x - 2) = √(2x + 3) - √(4x - 1).

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Accepted Answer

Hello. Today we're going to be solving the following equation. So the following equation is the square root of eight x minus four equal to the square root of two. X plus one minus the square root of eight x minus two. Now, what we need to go ahead and do is open up the quantities from the square roots in order to do that. We can go ahead and square both sides of this quantity. Doing so is going to give us eight x minus four and we're going to have to foil the right hand side, foiling the right hand side is going to give us two X plus one minus two times the square root of two, X plus one times the square root of eight x minus two Plus eight X -2. And what we need to do is go ahead and combine like terms well two x and eight x can can be combined together and negative two and one can be combined. So what we get afterward is eight x minus four equal to 10 x minus one minus two times the square root of two X plus one times the square root of eight x minus two. Now we want to go ahead and get this quantity of the square roots by itself. In order to do that. We can subtract 10 x from both sides of this quantity and we can add one to both sides. Doing so is going to give us negative two X minus three is equal to the negative two times the square root of two X plus one times the square root of eight x minus two. And since all of these quantities are negative, we can go ahead and divide everything by negative one. To change everything a positive So doing so is going to give us two X plus three equal to positive two times the square roots. Now, again, what we need to go ahead and do is square both sides of this quantity because you want to try to open up the square roots. Doing so is going to give us two x plus three squared, which is going to give us four X squared plus 12 X plus nine equal to four times the square root of two X plus one times the square root of eight X minus two squared which is going to give us four times the quantity 16 X squared plus four x minus two. Now what we need to do is go ahead and distribute four into this quantity and doing so is going to give us four X squared plus 12 X plus nine is equal 2 64 X squared Plus X -8. Now we want to go ahead and move everything to one side of the equation and set this equation equal to zero because we're trying to sell for X. So in order to do that we can go ahead and subtract four X squared to both sides of the equation. Subtract 12 X to both sides and subtract nine to both sides and doing so is going to give us X squared Plus four X -17 is equal to zero. Okay, so we managed to go ahead and simplify our original equation. What we need to do now is solve this equation and we can go ahead and do that by using the quadratic equation. In order to use the quadratic equation. We need to go ahead and identify an A. B and C value while A is just going to be the coefficient in front of the X square term. So A is going to equal to 60. B is going to equal to four and C is going to equal to negative 17. So if we plug this into the quadratic formula, what we get is X is equal to negative B plus or minus the square root of B squared minus four times a times C. All of that over two times a and simplifying this while X is going to equal to negative four plus or minus the square root of four squared which is 16 -4 times 60 times -17. Which is going to give us positive And all of that's going to be divided by two times 60 which is going to be 120. So continuing this plus 4080 is going to give us 4096 so X is going to equal to negative four plus or minus the square root of All divided by 120 and the square root of 4096 can be evaluated to so X is equal to negative four plus or minus 60 for over 120 now what we need to go ahead and do is separate the statement here. So we get two instances of X. X is going to equal to negative four plus over 120 X is going to equal to negative four minus 64 over for the left hand side, negative four plus 64 is going to give us 60 so X is going to equal to 60 over 120 simplifying that X is going to equal to one half and on the right hand side at negative four minus 64 is going to give us negative 68 so X is equal to negative 68 over 120 And simplifying this, X is going to equal to negative 17/30. So this is the values of X that we end up with after the quadratic formula but we need to verify that both of these values work with our original equation. So if we were to take X is equal to one half and plug it into our original equation, we get the square root of eight times a half minus four is equal to the square root of two times a half plus one minus the square root of eight times a half minus two And eight times one half is going to give us four. So we are left with the square root of 4 -4 on the right hand side, two times a half is just going to give us one. So we have the square root of one plus one minus eight times a half, which is four. So we are left with the square root of four minus two, 4 -4 is going to give us zero. So we have the square root of zero and the square root of one plus one is going to be the square root of two minus the square root of four minus two, which is going to give us the square root of two. So the square root of zero evaluates to zero and the square root of two minus the square root of two is equal to zero, which is a true statement. So that means that x is equal to a half, is going to be one of our correct solutions And we need to go ahead and check when X is equal to negative 17/30. Well this is going to be a pretty complicated number to work with but think about it this way, this is a negative number. And if we plug it into our original equation, we have eight times a negative number minus a number. So we're just going to end up with a larger negative number. And we're not allowed to have a negative number inside the square root because it breaks the domain. So if we were to plug in negative 17/30 to our original equation, it's going to cause an undefined value to occur. So we're not gonna use negative 17/30. That means that the only solution to this original equation is going to be X is equal to a half. And that means that the solution to this problem is going to be be so I hope this video helps you in understanding how to solve this type of equation and I'll go ahead and see you all in the next video.

Solve: √(6x - 2) = √(2x + 3) - √(4x - 1).

Key Concepts

Square Roots

Isolating Variables

Squaring Both Sides

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