Question 1209239: what number multiplies to get -432 and adds to get 6
Found 4 solutions by math_tutor2020, greenestamps, ikleyn, josgarithmetic:
Answer by math_tutor2020(3817)   (Show Source):
Answer by greenestamps(13200)  (Show Source):
You can put this solution on YOUR website!
This is one of those problems where the standard formal algebraic approach doesn't help.
The two numbers add to 6, so represent them with x and (6-x). Given that their product is -432,
To solve that using the standard algebraic approach by factoring, we need to find two numbers whose difference is 6 and whose product is -432. But that is what the original problem required us to do, so that work gets us no closer to the answer.
The response from the other tutor says we could solve the problem by trial and error but that using the quadratic formula is the easier way.
I think solving the problem by trial and error gives the student much more good brain exercise than plugging numbers into the quadratic formula.
Ignoring the signs for the moment, we need to find two integers whose product is 432 and whose difference is 6. Do this by finding the prime factorization of 432 and finding a way to combine those factors into two integers whose difference is 6.
432 = 3*144 = 3*12*12 = 2*2*2*2*3*3*3 = 16*27
The difference between the two integers we are looking or is 6, and the prime factorization contains several factors of 2. That means both of the numbers we are looking for must be even.
Playing around with the factors then leads us to (2*2*2*3)*(2*3*3) = 24*18.
Now, paying attention to the signs of the two numbers, since the sum of the two numbers is 6, the two numbers have to be 24 and -18.
ANSWERS: 24 and -18
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And now here is a way to get the answer using a nice formal algebra "trick".
We again ignore the signs for the moment and look for two numbers whose difference is 6 and whose product is 432.
Since the difference between the two numbers is 6, we can represent them with (x+3) and (x-3). Then, with their product being 432, we have
And the two numbers we are looking for (still ignoring the signs) are x+3 = 21+3 = 24 and x-3 = 21-3 = 18.
Then, as in my earlier discussion, we pay attention to the signs and find that the two numbers we want are 24 and -18.
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In that second solution, I initially ignored the signs of the two numbers, because that is easier for me to understand. You can also solve the problem using the same algebra trick without ignoring the signs, as follows.
Since the two numbers add to 6, represent them using (3+x) and (3-x). Then, with their product being -432,
And the two numbers we are looking for are 3+x = 3+21 = 24 and 3-x = 3-21 = -18.
Answer by ikleyn(52781)  (Show Source):
Answer by josgarithmetic(39617) (Show Source):
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