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Answers
If 52−x+x−5x+2+3x+8x2−4=0 , what is he value of x ?
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The first thing to notice is that the denominators may equal 0 .
So the values x∈{−2,2} require extra attention. To be safe, one should exclude these values entirely.
The next thing to notice is that x2–4=(x−2)(x+2) , actually I used this already.
Now if the denominators where numbers having a similar property, like:
12+73+56 what would be the next step?
You can't add fractions if the denominators are not equal. So you probably learned that one must first make sure they are equal, multiply the numerator and denominator with the same number and make sure that all resulting denominators become equal:
12⋅33+73⋅22+56 or
1⋅32⋅3+7⋅23⋅2+56 or
36+146+56
Since the denominators now are equal, we can add the parts of the pie, since we slice each pie into parts of equal size 1/6 .
We apply the same trick here, compare the denominators:
2 with (x−2) , 3 with (x+2) , 6 with (x−2)(x+4)
The only problem is that the first fraction in the question has a slightly different denominator 2−x . But that can be repaired:
52−x=−5x−2
Do you see that we applied the same trick as in:
2−3=−23 ?
Now we know what to do!
Change the first fraction to give it the correct denominator x−2
Multiply the first fraction with x+2x+2
Multiply the second fraction with x−2x−2
Leave the last fraction as is, the denominator is just fine as is.
We don't have to calculate the products that appear at the denominator, since we know that the answer should be (x−2)(x+2) .
Just as with numbers, we may finally add the numerators.
If you do your calculations correctly you should end up with:
x2–9x+8x2–4=0 or
x2–9x+8=0
Where we used that a/b=0,b≠0 gives a=0 .
I assume you know how to solve this equation.
Do check that the values you come up with are not equal to −2 or 2 .
Answer:
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