A couple of notes on the previous responses.
windbreaker;339051 wroteAnswer can't be x=2. When you solve for x, should plug it into the original equation to check your work. If it's correct...the math should work...in this case 24=24.
If it's not clear a = 3 b = 4 c = -28
Then you plug it into the quadratic equation to solve for two possible answers for x. Sometimes you don't get real answers as one or both answers may involve the sqrt of a negative number.
Quadratic equation - Wikipedia, the free encyclopedia
If you use the quadratic formula, you can get one or two real roots or two imaginary roots (if the value calculated under the square root - known as the discriminant - is a negative number). It is not possible to get just one imaginary root.
moose;339050 wrote
then I usually google for an online quadratic solver which uses the Ax^2+Bx+C=0 format, though most decent calculators nowadays have it preprogrammed.
I'm assuming that the teacher of the class isn't going to allow your daughter to use Google, or possibly even a calculator that is programmed with the quadratic equation during a test.
My guess is that in fact the teacher expects your daughter to factor the equation.
You started with:
(x+4)(3x-2)=24
Simplify and make one side of the equation 0. As Moose suggested, using FOIL works.
3x^2 - 2x + 12x - 8 = 24
3x^2 + 10x - 32 = 0 (combine like terms and subtract 24 from both sides)
I think they teach tricks in school now for the next part, but I didn't learn them myself. You can check your daughter's notes to see if she has something that looks like a table of factors of the first term and last term's cooefficients - in this case 3 and -32.
So I am looking for a combination of two factors of 3 and two factors of -32, that when I multiply one factor from each and then add those products I will get 10 (the coefficient of the middle term). I look at this I see that 1*16 - 2*3 = 10. The factors of 3 I used (1 and 3) go before the x's, in the first terms of the factored equation, and the factors of -32 (-2 and 16) are the second terms of the factored equation. (Note that I considered the other factors of -32, like, 4 and -8, -4 and 8, -16 and 2. I think if your daughter has been taught a procedure for this kind of factoring, these numbers would be part of that.) So the factored equation is:
(x - 2)(3x + 16) = 0
You can use FOIL again to make sure this has been correctly factored. This means that either (x - 2) = 0 or (3x + 16) = 0, since 0 times anything is 0.
If (x - 2) = 0, then x must be 2 (solve a linear equation). If (3x + 16) = 0, then x must be -16/3. However since we are talking about lengths, then (x+4) must be a positive number and (3x-2) must be a positive number. In the case where x=-16/3, these values would be negative. Therefore, the only possible answer is that x = 2.
