Solve without squaring both sides:

5 - (2/x) = √(5 - 2/x).

Question

Solve without squaring both sides:

5 - (2/x) = √(5 - 2/x).

Accepted Answer

Hello today we're going to be solving the following equation without squaring. So the equation given to us is 7 -4 over X is equal to the square root of 7 - over X. Now in order to solve this without squaring. What we can go ahead and do is use a substitution. We can substitute 7 -4 over X because it's present on both sides of this equation. So if we let a variable y equal to 7 - over X that's gonna cause our equation to be why is equal to the square root of why? Now there are only two numbers that make the statement true and that is when y is equal to zero and y is equal to one. So since we know that these two values are the only values that work with this statement here. What we can go ahead and do is take our values of Y and plug it into our substitution in order to figure out what our values of X are going to be. So let's go ahead and do that. And we can start when y is equal to zero. When y is equal to zero, plugging that into our substitution is going to give us zero is equal to seven minus four over X. In order to solve this, I can go ahead and add four over X to both sides of the equation. Doing so is going to give me four over X is equal to seven and I want to go ahead and get this X out of the denominator in order to do that. I can multiply both sides of this equation by X. And doing so is going to leave me with four is equal to seven X. And finally I can divide both sides of this equation by seven to give me X is equal to 4/7. So that's the first value of X. Now we need to figure out the value of X when y is equal to one. When Y is equal to one, we get one is equal to 7 -4 over X. In order to solve this, we can go ahead and subtract seven from both sides of this equation And doing so is going to give me -6 is equal to negative four over X. And multiplying both sides by negative X is going to give me six, X is equal to four. And if I divide both sides of this equation by six, I get X is equal to 4/6, which reduces to 2/3 and that's the second value of X that we end up with. Now we need to go ahead and verify that both of these solutions of X work with our original equation. So we need to go ahead and plug in our values of X to the original equation. So when X is equal to 4/7 we get seven minus 4/4 over seven is equal to the square root of seven minus 4/4 over seven and we need to go ahead and simplify 4/4 over seven is a complex fraction. And in order to get rid of the complex fraction, we can multiply both the numerator and denominator by the reciprocal of the denominator. So we're gonna go ahead and do that to both sides of the equation. So once again we're multiplying this by 7/4 and 7/ and doing so it is going to leave me with seven minus well 4/7 times 7/4 is going to cancel it to give us one. So these denominators cancel out and we're left with 7/ times four, both of the fours are gonna cancel out to leave us with just seven. So we have 7 -7 on the left hand side of the equation and the square root of 7 -7 on the right hand side of the equation, seven minus seven gives us the value of zero. So we have zero is equal to the square root of zero and the square root of zero is just zero. So we have zero is equal to zero which is a true statement. So the value of X is equal to 4/ is one of our solutions. Now we need to go ahead and check when X is equal to 2/3, so when X is equal to 2/3, we get seven minus 4/2 over three is equal to the square root of seven minus 4/2 over three. Now, just like before, in order to simplify the complex fraction, we're going to multiply both the numerator and denominator by the reciprocal of the denominator. Alright. And doing so it's just gonna cancel out both of the denominators in the complex fraction and four times 3/2. Well four and two are going to cancel out leaving us with two times three. So what we are left with on both sides Is 7 -2 times three, which is 7 -6, is equal to the square root of 7 -6 And 7 -6 gives us the value of one. So we have one is equal to the square root of one and the square root of one is just one. So we are left with the statement, one is equal to one, which is a true statement and that means that X is equal to 2/ is our other solution to this equation. So our solutions to the original equation are X is equal to 4/7 and X is equal to 2/3. With that being said, the solution to this problem is going to be a so I hope this video helps you understanding how to solve this equation without squaring and I'll go ahead and see you all in the next video

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