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

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Accepted Answer

Hello, everyone in this video, we're going to be looking at this practice problem where we want to solve for X and the equation, the square root of four, X minus three is equal to five plus the square root of X minus five. So in this case, we're solving for X, which means we're going to try to get all the X variables to one side because right now you can see that we have X on both the left hand side and the right hand side. So in order to start to move back more easily, I'm going to take the square of both sides, which means I'll just square both sides. And when I do that, the left hand side becomes four, X -3. And my right hand side is now five plus the square root of X - times five plus the square root of X minus five. I'm gonna go ahead and multiply the right hand side. So I have five times five is 25 times the square root of X plus five is five times the square root of X minus five. And then we have the square root of X -5 times five, again, 5 times The square root of X -5. And then we have the square root of X -5 times the square root of X minus five. And that gives us X minus five. So we can drop the square root. So now we're left with four, X minus three on one side is equal to 25. We can combine the middle terms to leave us with 10 times the square root of X minus five plus X minus five. And we can actually subtract five from 25 leaving us with 20. And my next step from here is going to be to move this X over to the left hand side by subtracting X from both sides. So my left hand side becomes three, X minus three. They'll actually also subtract this 20 from both sides. So negative 3 -20 is negative 23 And that's equal to 10 times the square root of X -5. And once again, you can see that we have an X in a square root. So I'm going to take the square one more time of both sides. So now my left hand side is three X - times three X -23. And that's equal to 10 squared And we can drop the square root. So we're 10 squared times X -5. So I can expand the left hand side. Once again, I have three X times three X is nine X squared three X times negative 23 negative 69 negative 23 times three X again is negative forgot six here negative X and again, we have the same multiplication negative 23 times positive three X is negative 69 X And then finally negative times negative 23 Is positive 529. That's equal to 10 squared is 100. If I distribute it, I have 100 X minus 100 times negative five is negative 500. And I can move all of these to the Left hand side. So I'll subtract 100 x and add 500. And in order to make this, In order to add these two components easier, I'll combine the middle terms here. So negative 69 minus negative of 69 x -69 x is 100 -138 X. And then we have this minus 100 X. So this becomes nine X squared minus X and then positive plus 500 is Positive 1029. And the right hand side is now zero. So from here, you can see that we have a quadratic equation. So I'm going to figure out the values for X by factoring. So for my factoring, the first terms need to multiply 29 X squared. So when I think of terms that multiply to nine X squared, I could have three X times three X or nine X times one X and the negative versions of those as well. For now, I will try nine x and one x and then the last terms need to Multiply 2, 1029. So some factors that multiply to 10:29 is 49 and seven and 1 three and 3 43. I'm just gonna go ahead and try these because they are smaller numbers to work with. So say nine Plus 49 and not one x plus 21. And I can go ahead and test these by doing a quick foil. So nine X times one X is nine X squared, nine X times 21 is 89 X And then 49 times one x is 49 x and 49 times is 29. I combine these middle terms, I get nine x squared plus Plus 238 X plus 1029. And from here, you can see that this middle term has the wrong sign. We wanted it to be negative, not positive. So I can go back to my factory ization and because I want the middle term to be negative, but my last term to still be positive, I'm going to do negative 21 negative 49. You can see when I multiply negative 49 times negative 21 I still get positive 1029. But now when I do this multiplication, I have nine X times one X is still nine X squared nine X times negative 21 is now negative 189 X and negative 49 times one X is now negative 49 X and then negative 49 times negative 21 is still positive 1029. Which means that now when I combine these middle terms, I get B -238 that I was looking for. So that I confirm that my factoring is correct. I can solve for X in each of these factors. So I have nine x -49 is equal to zero and 1 X -21 is equal to zero. So if I solve for X In this first scenario at 49 to both sides Means nine, X is equal to All divide by nine, That means X is equal to 49 divided by nine. So this is one of my possible solutions. We'll do the same on this side. So I'll add 21 which means X is equal to positive 21. So this is a second solution. And you may be thinking, well, these are the two solutions. Those are the ones we need to choose. Well, we still need to check and make sure that they work in our equation. So our original equation stated that the square root of four x minus three was equal to five plus the square root of X minus five. So now we can test both of these values that we found. So I'll start by testing X equals 21 1st. So when X is equal to 21 my equation is the square root of four times 21 minus three is equal to five plus the square root of 21 minus five. So four times 21 is 84 -3 is still under a square root and the square root of 81 is Plus or -9 and that is equal to the right hand side which we can also simplify. So five plus 21 minus five is equal to 16. The square root of 16 is four. So five plus four, We'll get nine on both sides. So x equals 21 works with our original equation and now we can try x equals 49 divided by nine. This becomes four times 49 divided by 9 -3. Still taking the square root Is equal to five plus the square root of 49 divided by 9 -5. So we have four times 49/9. So multiply four times 49 is 96/9 -3. I could write this as 27/9 27/9 is still three just with a common denominator. So I can subtract it from this fraction here. So 196 -27 is equal to 169 divided by nine still in the square root. And I'll look at the right hand side. So I have five plus again, I'm going to write this five with the denominator of nine. So five times nine is 45 over nine. So this fraction is still five. Now with the common denominator, so I can combine the two have the square root of 49 -45 is 4/9. So I can take the square root of 4/9. This becomes five plus the square root of four is 2 and the square root of nine is 3. And I can also take the square root of 169. 13 in the square root of nine is 3. If I write five with a denominator of times three is 15 divided by three. So I have 13, divided by three is equal to 17, divided by three. And that is not true. So x equals 49/9 is not a solution, which means that we only have one solution Where x equals 21. And if we look at the answer choices, you can see that answer choice B list 21. So hopefully this video helped and see you next time.

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

Key Concepts

Square Roots

Isolating Variables

Extraneous Solutions

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