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Solving Quadratic Equations in Disguise - Problem 2
University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
So hopefully you remember how to factor out an equation when you’re dealing with negative exponents but if you don’t it’s really quite simple, all you have to do is factor out the largest negative. So the smallest exponent you’re dealing with.
First thing we want to do when solving this out is factor out our smallest number, our largest negative, in this case we’re looking at x to the -5. We factor out a x to the -5 and then we need to fill in the blanks. Remember when we are multiplying, we want to figure out what goes into this first term. Remember when you multiply bases you add exponents so -5 plus something is equal to -3. This is going to turn into x², minus 9.
Now we need to go through the same process to find out what plus -5 is -4, that’s just going to be to the first, so that’s just an x and lastly we end up with plus 20 is equal to zero.
So by factoring out our smallest negative we were actually able to turn this fairly ugly statement to a quadratic which we now know how to solve. Factoring this out, we end up with the x to the negative 5th stays on the outside, x minus 5, x minus 4 is equal to zero. Whenever we’re multiplying things together to equal zero, one or more have to equal zero so this tells us that we have, 4 is zero, 5 is zero and let’s just take a second to interpret what this means.
Remember that negative exponents mean you flip your fraction so this is actually the same thing as 1 over x to the 5th. We want to figure out 1 over x to the 5th is equal to zero. The numerator is always 1 so the only thing that we could actually change is the denominator. 1 over zero is undefined, 1 over a number is always just going to be a number so this will actually never be equal to zero. So this will never give us a solution leaving the only possible answers being x is equal to 5 or x is equal to 4.
Solving an equation with negative exponents, just factor out that negative exponent and turn it into a quadratic.
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