In Exercises 1–38, solve each radical equation.
_____
x = √3x + ...

Question

In Exercises 1–38, solve each radical equation. 
_____
x = √3x + 7 - 3

Accepted Answer

Welcome back. I am so glad you're here. We are told in the given radical equation to solve for X. Our given radical equation is X equals the square root of three X plus 25. And then outside of the radical minus nine, our answer choices are answer choice A, the solution set negative eight, negative seven answer choice B, the solution set negative 87 answer choice C the solution set negative 78 and answer choice D, the solution set 78. All right. In order to solve for X, typically, what we want is we want X by itself on one side of the equation and everything else on the other side. And we almost have that here. We have an X equals on the left, but we have an X on the right hand side as well. And it is trapped underneath that radical. And whenever an X is trapped inside of a radical, our first step is to rescue that X. So in order to rescue the X from the radical, we need to isolate the radical on one side of the equation. Meaning anything that's not in the radical needs to move over to the other side. In this case, that's that minus nine. So we need to do the opposite operation. The nine is being subtracted. So we will add, we'll add nine to both sides of the equation. On the left, we'll have X plus nine that's equal to on the right, we have the square root of three X plus 25 and then that negative nine plus nine, that's zero. So we have isolated the radical on the right hand side of the equation. Great. Now how do we get rid of that radical? Well, when we're taking the square Rutt, we are going to square it. And what we do to one side, we need to do to the other as well. So on the right, the square root of three X plus 25 squared will simplify +23 X plus 25 that X is thrilled. It has been freed. But on the left, the quantity of X plus nine squared does need to be foiled out. So we'll take the quantity of X plus nine multiplied by the quantity of X plus nine. We multiply our first X multiplied by X will be X squared. Our outsides X multiplied by nine is plus nine X insides nine multiplied by X is plus nine X and lasts nine, multiplied by nine is plus 81. Simplifying, we can bring down the X squared combining the like terms in the middle nine, X plus nine X is plus 18 X and then bring down plus 81. So that's the left hand side of our equation simplified we have X squared plus 18 X plus 81. And now is where we want to get the XS by themselves and everything else on the right. However, we have here a quadratic, we have this X squared and then we have an X to the first power as well. So when we have a quadratic, we want to get all of the numbers on one side and set the other side equal to zero. And what I like to do is have the X squared, the highest degree be positive. So I am going to subtract three X from both sides of the equation to bring the X values from the right side over to the left. And then I'm going to subtract 25 from both sides of the equation in order to bring the constant terms from the right hand side over to the left. So on the left, we still have X squared, then we have 18 X minus three X which will be plus 15 X. And then we have a positive 81 minus 25 which is going to be plus 56. And on the right hand side of the equal sign, the three X minus three X is zero and the 25 minus 25 is zero. The right hand side equals zero, just what we wanted. Now, we can attempt to factor the left hand side. We'll do the reverse of the foil process. We'll set up two open sets of parentheses equal to zero. And we start off asking ourselves what multiplied by what equals X squared. Those are our firsts and that's going to be X multiplied by X. Then we ask ourselves what multiplied by what equals a positive 56. Those are our lasts. Keeping in mind that when adding our outsides plus our insides, they need to add up to a positive 15 X. So in other words, what two factors of positive 56 add up to positive 15. That's going to be a positive seven and a positive eight. And now we know that if either of those factors are zero, this equation will be true. So we set each of those factors equal to zero X plus seven equals zero and X plus eight equals zero. Now we solve for X for the first one, X plus seven equals zero, we subtract seven from both sides of the equation and we have X equals negative seven because the seven minus seven is zero on the left and zero minus seven is negative seven on the right and four X plus eight equals zero, we will likewise subtract eight from both sides of the equation and eight minus eight is zero and zero minus eight is negative eight. So we have X equals negative eight and we have solved four, X our solution set is negative seven, negative eight. We look at our answer choices and that matches with answer choice. A well done. We'll catch you on the next one.

In Exercises 1–38, solve each radical equation. _____ x = √3x + 7 - 3

Key Concepts

Radical Equations

Isolating the Variable

Extraneous Solutions

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