Unit
Quadratic Equations and Inequalities
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!
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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 the first step I always do when I’m solving a quadratic equation is to get everything to one side. In general I choose the side where my x² coefficient will be positive. In this case it’s going to be over here on the left.
So bring our 8x around, we’re going to put x² minus 8x plus 1 is equal to zero. We have a number of different ways to solve quadratic equations out. We could factor which is going to be hard given that we are dealing with 4 and 1 and trying to make -8, so that’s out. We could use our square root property but we don’t have any quantity squared so that’s out as well leaving our only two options left as completing the square or the quadratic formula. Either one is going to work for this case, in general, the quadratic formula tends to be easier for people because you’re just plugging in numbers. But if you wanted to complete the square it would work just as well.
Let’s just solve this out then by using our quadratic formula, which is, hopefully you remember, -b plus or minus the square root of b² minus 4ac all over 2a. Some people know jingles to remember it, I’m pretty boring, and I just remember it straight up. All we have to do now is plug in our numbers.
Our b is -8 so when I plug in the -8, the negatives cancel leaving us with 8 plus or minus giant square root, b² is 64 minus 4, a is 4, and c is 1, all over 2a, so all over 8. 64 minus 4 times 4, 64 minus 16, 64 minus 14 is 50 so that takes us down to 48. So what we’re left with is 8 plus or minus the square root of 48 over 8. 48 is 16 times 3, do we can take out a 4. Let’s come over this way and what we’re left with then is 8 plus or minus, we took out the 4 so this is 4 root 3 all over 8.
And we can simplify this up a little a bit more, two different ways we can simplify it; this 8 can go to both things, so if you wanted to divide everything by 4 we could be left with 2 plus or minus root 3 over 2, or if you wanted to split it up all together what we’d end up with 1 plus or minus root 3 over 2. Either one is fine, just different ways of interpreting this fraction.
So all we’ve done is we pulled everything to one side, realized we couldn’t factor it so we decided to use the quadratic formula and then simplified our quadratic formula down as much as we could.