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But you have to be very careful there because when you square radical signs you actually lose the information that you were taking the principal square root. So the first thing I want to do is I want to isolate this on one side of the equation.Not the negative square root or not the plus or minus square root. And so when we get our final answer, we do have to check and make sure that it gels with taking the principal square root. And the best way to isolate that is to get rid of this 3.
And this is where we actually lost some information because we would have also gotten this if we squared the negative square root of 5x plus 6. And now, this is just a straight up linear equation. So we need to make sure it actually works for the positive square root, for the principal square root. So we get 3 plus the principal square root of 5 times 15.
And so that's why we have to be careful with the answers we get and actually make sure it works when the original equation was the principal square root.
Inverse operations are critical for solving algebraic equations. The method for solving equations containing radicals involves the same process. Solve for x: The first step to solving is to remove the radical symbol by performing the inverse operation, which is to square both sides. When you square the left side, the square and square root cancel each other out, so you are left with x - 1. There is always the possibility that you will get an answer that is called an extraneous root.
For example, if you want to solve the following: x 2 = 3. To do that, you perform the inverse operation, to move the 2 to the right side of the equation. In order to isolate the x (or whatever the variable happens to be), you need to perform inverse operations to move all the numbers to the right side of the equation. That is just the most conventional way of doing things. This is an answer that seems to work, but isn't right when you check your answers.
Otherwise, I would lose the ability to say that they're equal. And now, we can square both sides of this equation.
And so the left-hand side right over here simplifies to the principal square root of 5x plus 6. So we could square the principal square root of 5x plus 6 and we can square 9. Or we get 3 plus square root of 75 plus 6 is 81 needs to be equal to 12.
And the best way to get rid of the 3 is to subtract 3 from the left-hand side.
And of course, if I do it on the left-hand side I also have to do it on the right-hand side.
We're asked to solve the equation, 3 plus the principal square root of 5x plus 6 is equal to 12.
And so the general strategy to solve this type of equation is to isolate the radical sign on one side of the equation and then you can square it to essentially get the radical sign to go away.