Radical Equations And Problem Solving

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Warning: Many instructors do not to show many examples (in class or in the homework) of radical equations for which the solutions don't actually work.

But then they'll put one or more of these on the next test.

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.

So I'd have been checking my solutions for this question, even if they hadn't told me to.

I'll treat the two sides of this equation as two functions, and graph them, so I have some idea what to expect. This is for my own sense of confidence in my work.) I'll graph the two sides of the equation as: solution. It came from my squaring both sides of the original equation. I can see it in the squared functions and their graph: ("Extraneous", pronounced as "eck-STRAY-nee-uss", in this context means "mathematically correct, but not relevant or useful, as far as the original question is concerned".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.(Yes, this means that you can use your graphing calculator to help you check your work.) When I was solving " lines had not intersected.This illustrates why I had to check my solution to figure out that the real answer was "no solution".My check is done by plugging the proposed solution into both the left-hand side (LHS) and right-hand side (RHS) of the original equation, and confirming that each simplifies to the same value (or else showing that the solution isn't any good): Even if the instructions hadn't told me to check my answers, clearly I needed to.And I needed to do that check with algebra, not with the picture.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.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.


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