If a number is decreased by 3, the principal square root of this ...

Question

If a number is decreased by 3, the principal square root of this difference is 5 less than the number. Find the number(s).

Accepted Answer

Hello. Everyone in this video, we're going to be looking at this practice problem where we're trying to find the number in the word problem and in this case we're given that if the number is decreased by 11, the positive square root of the difference is 13 less than the number. So let's go ahead and try to translate some of these clues into an equation so we can solve for that number. So in this case, wherever it says a number, because we're trying to solve for the number I'm going to call or do note my number X. So now I'll be solving for X. And this first part is saying a number is decreased by 11, So X -11 is decreasing the number by 11. And then it says the positive square root of the difference. So the positive square root of X -11 is 13 less than the number. So that they're saying that the square root of X -11 is equal to X -13 or 13 Less than the number. So now I've written what they've said in the word problem in a sort of equation that I can solve. So I'm going to go ahead and try to solve for X. And if I do this, I'll square both sides. So my left hand side becomes X -11 And my right hand side becomes X -13 times X -13. And if I continue this multiplication on the right hand side, I have x times X X squared x times negative 13, negative 13 X negative 13 times X negative 13 X and negative 13 times negative 13 or positive 169. And my left hand side is still X -11. So you can see that my right hand side is starting to look a lot like a quadratic equation. So I'm going to try to move everything into that quadratic equation so I can factor it it and have solutions for X. So I'm going to add 11 and subtract X from both sides. So that leaves me with zero on the left hand side And my right hand side becomes x squared, I have negative 13 minus 13 X. And another negative X. So this becomes negative 27 X. And then I have positive 169 plus 11, so plus 180. So now I have a quadratic equation and I'm going to factor it up. So remember when we're factoring out the quadratic equation, we want the numbers to multiply to 1 80 but add to 1 to 27 negative 27. So in this case if I use and multiply that by 12 I get 180. So let's see if I add 15 and I get positive 27 and I need negative so if I change that to negative 15 -12 now I have negative 27 And if I multiply negative 15 by negative 12 I still get positive 1 80. So now 15 and 12 become the numbers I will use for the factoring so x squared minus 27 X plus 1 80 becomes x -15 times x -12. And because we moved everything to one side, this is equal to zero. So now I can solve for X and it seems like there's two solutions, So X -15 is equal to zero, gives me X is equal to positive 15 And X -12 is equal to zero, gives me x equal to 12. So now it seems like the answers are, X is equal to 15 or X is equal to 12. However, let's go ahead and check that because it may seem like the answer is C. But if we look at X is equal to and we plug that into the equation that we arrived at or the condition that needs to be true, we have 12 minus is equal to 12 - and if we solve that we have The square root of one is equal to negative one, which means one is equal to negative one And that is not true. So let's just go ahead and check x at 15 to see if that makes this Equation true, we have the square root of 15 -11 equal to 15 -13. And if we do that we have the square root of four equal to two And two is equal to two. So you can see that X equals equal to 15 is actually the true answer. So whenever you're solving for these weird problems and you have a quadratic equation, it's important to make sure that both answers are correct because maybe only one of them Would make the condition true. So in this case you can see that answer choice a is also X equals 15, so hopefully this video helped and see you next time.

If a number is decreased by 3, the principal square root of this difference is 5 less than the number. Find the number(s).

Key Concepts

Algebraic Expressions

Square Roots

Solving Equations

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