If 5 times a number is decreased by 4, the principal square ro...

Question

If 5 times a number is decreased by 4, the principal square root of this difference is 2 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 need to determine a number based on the following clues. Six times the number is increased by three. The principal square root of the sum is four less than the number. So in this case we're trying to find the numbers that could fall under these conditions. So I'm going to represent the unknown number as X. So wherever I see number in my word problem I'm just going to say that that is X. So I'm trying to find the values for X. So this first part is saying six times a number. So six times X is increased by three. So increasing the number by three means adding three and then the principal square root of that some. So the principal square root of six x plus three is so equals for less than the number for less means I'm subtracting four from the number. So I have negative four. Or I write this out X -4 because I'm subtracting four from the number. So you can see that we're working here with an equation that has X values on both sides. I'm gonna now use this equation to simplify and try to get the value for X. So the first thing I'm going to do is square both sides to get rid of the square root. So that leaves me with six X plus three is equal to x minus four times X minus four. And I'm going to carry out this multiplication on the right hand side. So I have X times X. Is x squared negative four times X is negative four X. X times negative four is negative four X. And negative four times negative four is positive 16. And you can see that on my right hand side. I now have a quadratic equation where I have an X. Squared and then X. Value and then a constant. So I'm going to try to use the fact that I have a quadratic equation to try and solve for X. So I'm going to move what's on my left hand side six X plus three into this quadratic equation. So I'm going to subtract by six sex and subtract by three. And if I do that my quadratic equation becomes X squared and then I have negative four X minus four X. So negative eight X minus six. X. So negative 14 X. And then I have positive 16 -3 That becomes positive 13. So now my quadratic equation is X squared minus 14 X plus 13. So I'm going to factor this quadratic equation. And remember when I factor this quadratic equation, I want the two numbers to multiply 2, 13 and I want them to add to negative 14. So if I try 13 And one they multiply to 13. So that's good. But if I do 13 plus one I get positive 14 and I want negative 14. So now I'm going to change the signs to negative 13 and negative one. And that gets me negative 14. And if I multiply by negative 13 and negative one I get positive 13. Still so negative 13 and negative one become the numbers I'm gonna use two factor the equation. So my equation X squared minus 14, X plus 13 becomes X - times x -1. So now I'm gonna use these two terms to solve for X. So I have X -13 is equal to zero So X is equal to positive 13 and then I have X -1 is equal to zero, so X is equal to one. So in this case it seems like the answer is X is equal to 13 or X is equal to one. But let's go ahead and verify that to make sure that both of those solutions work for the equation that we built. So recall that the equation that we're testing is six, X plus three Is equal to X -4. So we want to make sure that when X equals 13 or X is equal to one, that this equation is true. So let's try at X equals 13. We have the square root of six times 13 plus three equal to x -4. So 13 -4. If we solve this out, we have six times 13 or 78 plus three is equal to nine or the square root of 81 is equal to nine, which means nine is equal to nine and this is true. So x equals 13. Our equation is true. Now let's check out X is equal to one. And if we do that We have the square root of six times 1 plus three is equal to one minus four. So we have the square root of six plus three is equal to negative three or the square root of nine is equal to negative three. So we're saying three is equal to negative three and you can see that that's not true. So in this case the only solution that satisfies our equation is x equals 13. So that becomes our final answer. So if you stopped early enough and didn't verify the answer, C. Would have maybe been the answer he chose. However, as we've seen, X equals one, does not work. So only X equals 13 satisfies the equation. So hopefully this video helped and see you next time.

If 5 times a number is decreased by 4, the principal square root of this difference is 2 less than the number. Find the number(s).

Key Concepts

Formulating Algebraic Equations from Word Problems

Square Roots and Principal Square Root

Solving Quadratic Equations

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