(x-y)/(x+y) + (x+y)/(x-y)
Since you are adding, you find a common denominator. The common denominator in this case is (x+y)(x-y). We need to multiply each fraction to get this as a denominator. Remember to do the same thing to the numerator that you do to the denominator. When you do this, you get:
(x-y)(x-y)/(x+y)(x-y) + (x+y)(x+y)/(x-y)(x+y)
You now foil the numerators, and you get this:
(x^2-xy-xy+y^2)/(x+y)(x-y) + (x^2+xy+xy+y^2)/(x+y)(x-y)
combine like terms on top, and FOIL the bottom. When you do, you get this:
From here you should be able to figure it out.
2. Melissa was correct in pointing out that you were supposed to subtract 10 from both sides, not add. The result is a trinomial that looks like this:
12x^2 + 7x - 10 = 0
This is expressed algebraically by substituting constants for the coefficients, like this:
ax^2 + bx + c = 0
If you have a leading coefficient (the 'a' in the equation) that is NOT 1, here is what you do:
Multiply the 'a' and 'c' coefficients together. In this case, the answer is -120.
Now you find the factors of 120 that, when added, equal our 'b' coefficient, which is +7, in our case. Here is the process, quickly:
"What times what equals 120?"
These are all the factors of 120. We know that one has to be a negative number, since our 'c' coefficient is negative. Looking at our list, only 8, 15 provides this for us. If we used -8 and +15, when we add them we get +7, and when we multiply them we get -120. This is what we are looking for.
(12x^2 - 8x) (+ 15x - 10)
We factor any common terms. Now the terms in the parenthesis should be the same. If they aren't, you did something wrong:
4x(3x - 2) 5(3x - 2)
Since the two sets in parenthesis match, as they should, we just use that set as one binomial:
(3x - 2)
And now we combine the two terms outside the parenthesis to make a new binomial:
(4x + 5)
Our new equation looks like this:
(3x - 2)(4x + 5) = 0
If the set within either parenthesis equals zero, it will multiply with the other set to make our equation 0=0. All we need to do is find out what value of x will result in a zero answer for one of our parenthesis sets. In order to do this, we take each binomial and equal it to zero, then solve for x. I will work the first one:
3x - 2 = 0
3x = 2
x = 2/3
If we plug this into the first binomial, then the binomial will equal zero, and when we multiply it by WHATEVER is in the other binomial, we will get zero, as well. Therefore, 2/3 is one of the solutions. There is one more. I will leave it to you to find.
3. That's what I got
4. That's what I got
Hopefully this helps! Please feel free to contact me if you have any questions. Thanks!