Solve each equation. See Examples 4–6. √(3√(2x+3)) = √(5x-6)

Question

Accepted Answer

Hello Today we're going to be solving for X in the given equation. So what we are given is the square root of five times the square root of six X plus seven is equal to the square root of six X minus seven. So what I want to go ahead and do is set the equation equal to zero in order to solve for X. But currently I want to remove the inside arguments of the square roots on both sides of the equation. In order to get rid of the square roots, I'm going to go ahead and square both sides of the equation and doing this is going to leave me with five times the square root of six X plus seven is equal to six x minus seven. Now once again I want to remove the inside argument of the current square root So I'm gonna once again square both sides of the given equation and doing this while on the left hand side I'm going to have to distribute this exponent of two to both five and the square root of six X plus seven. Doing so is going to give me five squared which is going to be 25 times the quantity six x plus seven and that is going to equal to six x minus seven squared. Now, in order to simplify the right hand side I'm going to need a foil, six x minus seven squared six x minus seven squared is the same thing as saying six x minus seven times six x minus seven. In order to foil this I'm going to multiply six X to six X. Which is going to give me 36 X squared Then multiply six X 2 -7 which is going to give me -42 x. Then I'm going to multiply negative 7 to 6 X. Which is going to give me 40 negative 42 X. Then finally multiply negative seven to negative seven which is going to give me positive 49 I can combine negative 42 X with negative X. And that's going to give me 36 X squared minus X plus 49. So this is what the right hand side of the equation is going to simplify to. So again foiling the right hand side is going to give me 36 X squared minus 84 X plus 49. And in order to simplify the left hand side, I'm going to distribute 25 into the quantity six x plus seven. And this is going to give me 150 x plus 175. Now again, I want to go ahead and set this equation equal to zero. So what I can go ahead and do is subtract 100 and 50 X to both sides of the equation And I can subtract to both sides of the equation. Doing this is going to give me zero is equal to 36 X squared - x -126. Now on the right hand side, I can go ahead and simplify the coefficients of the equation. If I divide everything by nine, what I am left with is zero is equal to four X squared minus 26. X minus 14. 14. Yeah, 14. And once again I can simplify this even further by dividing everything by two. And dividing everything by two is going to leave me with zero is equal to two, X squared minus 13 X minus seven. Now on the right hand side of the equation, I have a fact Herbal Tryin Amiel. This Tryin Amiel is going to factor to the quantities two X plus one times X minus seven and that's going to be equal to zero. Now finally, in order to sulfur X, I'm going to take both factors of X. Set them equal to zero and solve for X. And this is going to give me two, X plus one is equal to zero and x minus seven is equal to zero. On the right hand side. I can go ahead and add 7 to both sides of the equation and that's going to give me X is equal to seven. And on the left hand side I can subtract one to both sides. Leave me leaving me with two X is equal to negative one. And dividing both sides by two is going to give me X is equal to negative 1/2. Now I have both potential solutions for X but now I need to verify the solutions for X. In order to do that, I'm going to need to take X is equal to negative one half and X is equal to seven and plug it into the original equation. So let's go ahead and do that. Let's first check, X is equal to negative one half. Plugging this into the original equation is going to give me the square root of five times the square root of six times negative 1/2 plus seven. Now six times negative half is going to give me negative three and negative three plus seven is going to give me the square root of four. So on the left hand side of the equation I have the square root of five times the square root of four. The square root of four is going to evaluate to two. So what I have is five times two which is going to give me the square root of 10. Now on the right hand side of the equation that is going to equal to the square root of six times negative half minus seven. Now the problem here is that six times negative half is going to give me negative three and -3 -7 is going to give me negative 10. But what I have is negative 10 underneath the square root which is going to produce an imaginary value since we have an imaginary value in the equation X is equal to negative one half is not going to be a solution. Now let's go ahead and verify X is equal to seven. If we plug in X is equal seven we have the square root of five times the square root of six times seven plus seven is equal to the square root of six times seven minus seven. Now on the left hand side six times seven is going to give me 42 plus seven is going to give me 49. So what I have is the square root of five times the square root of 49 that's going to equal to six times seven which is minus seven which is going to give me 35. Now the square root of 49 is going to evaluate to give me seven and seven times five is going to give me the square root of 35 And what I am left with is the square root of 35 is equal to the square root of 35 which is a true statement. So that means X is equal to seven is going to be the only solution for the given equation. And with that being said, the answer to this problem is going to be a so I hope this video helps you in understanding how to approach this problem and I'll go ahead and see you all in the next video.

Solve each equation. See Examples 4–6. √(3√(2x+3)) = √(5x-6)

Key Concepts

Square Roots

Radical Equations

Isolating Variables

Watch next