Solving with the Quadratic Formula
The quadratic formula
You just passed level one of Furious Fowls, the game that asks you to launch birds across the screen to get back at those pesky pigs that stole your eggs. You learned that we could figure out where our bird would hit the ground by solving the given quadratic equation where it equals 0. That required us to factor the equation and then use the zero product property to determine which two values would give us 0.
But now we're on to level two! It's going to get harder, but your goal will still be the same: use the equation given to you on the screen to make sure that your bird will hit its target on the first try. Alright, so here we go; let's go ahead and make a guess at where we'll need to launch our bird this time. I think this looks about right. Now, let's check our equation to see if this will actually work out or not.
The game tells us that our bird will fly in a parabola given by the equation y = -x^2 + 4x + 7. Since we're again shooting for a pig on the ground, we're curious where this equation equals 0, so let's go ahead and substitute that in. Okay, now we need to try to factor in order to find x. Let's divide out the negative, leaving us with this: 0 = -(x^2 - 4x - 7). Now look for a pair of numbers that have a product of -7 and a sum of -4. Hmmm, this isn't looking good. I don't think this trinomial can be factored. So, what now? In level one this worked every time, so we never had to do anything else! Looks like we're going to have to learn some new skills.
The Quadratic Formula
Situations just like this, where we're trying to solve a quadratic that is unfactorable, is what the quadratic formula was made for! It's a little long and messy, but if you can remember it, you'll be able to solve any quadratic equation out there!
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The resulting problem for example #1 after dealing with exponents and multiplication
The Standard Form
You'll have to first remember the standard form of a quadratic equation, y = ax^2 + bx + c. This will tell you what your a, b and c values are, but then it simply becomes a matter of putting the numbers into the right spots. Let's see if we can successfully do this to pass the second level of Furious Fowls.
Identifying A, B and C
Going back to the original equation and identifying a, b and c should be our first step. A is in front of the x^2, where there's only a negative symbol. That means that a is just -1. B is the number in front of the xs, which makes it 4, and c is the constant on the end, 7.
Using a Negative B Value
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Resulting example #2 problem after substituting values into formula
One other common mistake I see students make is to mess up the b value in the front left of the fraction. In the formula, it's a -b, so in this case 15 turns into -15. But often times your b value itself will be negative. When you have a -b and you plug it into the formula, - -b will turn it positive again, so you've got to be careful with that b in front.
Anyway, moving on to evaluating, we can start with the exponent on the inside - the b^2 part of the discriminant - and 15^2 gives us 225. Next, we do 4 ×-3 ×29 to get -348, and we do 2 ×-3 to get -6 on the bottom. Again, changing a minus negative into plus a positive will make our discriminant equal to 573. The square root of 573 is about 23.937. Doing -15 plus this and dividing by -6 will give us -1.489, and doing -15 minus this divided by -6 will give us 6.489.
We can again focus on the positive answer even though the negative one is a valid mathematical solution, and it looks like this one's going to work! I'm feeling pretty confident; let's let our furious fowl fly... and hey-o! We got it! You've now passed level two of Furious Fowls and are pretty good at solving quadratics, but are you ready for level three?