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For most of this lesson, we'll be working with square roots.For instance, this is a radical equation, because the variable is inside the square root: In general, we solve equations by isolating the variable; that is, we manipulate the equation to end up with the variable on one side of the "equals" sign, with a numerical value on the other side.Yes, the picture can be extremely helpful, but it isn't proof.
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.
For instance, in my first example above, " Squaring both sides of an equation is an "irreversible" step, in the sense that, having taken the step, we can't necessarily go back to what we'd started with.
By squaring, we may have lost some of the original information.
When you do this-- when you square this, you get 5x plus 6. So we get x is equal to 15, but we need to make sure that this actually works for our original equation. And this is the principal root of 81 so it's positive 9.
If you square the square root of 5x plus 6, you're going to get 5x plus 6. On the left-hand side, we have 5x and on the right-hand side, we have 75. We get x is equal to-- let's see, it's 15, right? Maybe this would have worked if this was the negative square root. So it's 3 plus 9 needs to be equal to 12, which is absolutely true. The left-hand side of the equation can be graphed as one curve, and the right-hand side of the equation can be graphed as another curve.The solution to the original equation is the intersection of the two curves. If the instructions don't tell you that you must check your answers, check them anyway.At the very least, compare your solution with a graph on your graphing calculator.If the term hasn't come up in your class yet, you should expect to hear it shortly.) By squaring both sides, I created an extra (and wrong) solution.Now I'll prove which solution is right by checking my answers.The general process for isolation is, in a sense, undoing whatever had been done to the variable in the original equation.For instance, suppose we are given the following linear equation: We can always check our solution to an equation by plugging that solution back into the original equation and making sure that it results in a true statement.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.