SOLUTION: what numbers multiply to -7 and adds up to 10

Question 1207031: what numbers multiply to -7 and adds up to 10

Found 2 solutions by math_tutor2020, josgarithmetic:
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

Let's list out the ways to multiply to -7 using integers only
1*(-7) = -7
-1*7 = -7

But unfortunately none of the pairs add to 10
1 + (-7) = -6
-1 + 7 = 6

You'll need to use the quadratic formula to solve x^2+10x-7 = 0 to get these roots
x = -5 + 4*sqrt(2) = 0.656854
x = -5 - 4*sqrt(2) = -10.656854
The decimal values are approximate.

Flip the sign of each value to end up with: -0.656854 and 10.656854

Check:
-0.656854*10.656854 = -6.999997177316 approximately which is really close to -7
-0.656854+10.656854 = 10
The rounding error happens because we rounded the roots mentioned above.

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If you meant to ask "what numbers add to -7 and multiply to 10?", then the two numbers are -2 and -5.
This is found through trial and error.

It might help to list all of the possible ways to multiply to 10 using integers
-1*(-10) = 10
-2*(-5) = 10
1*10 = 10
2*5 = 10
The second row is what we want since -2 + (-5) = -7

Notice how these two values help us factor x^2-7x+10 into (x-2)(x-5)
You can check this using the FOIL rule or the box method.

Answer by josgarithmetic(39617) (Show Source): You can put this solution on YOUR website!
xy=-7,x+y=10

y=10-x
x(10-x)=-7
10x-x^2=-7
-10x+x^2=7
x^2-10x=-7
x^2-10x+7=0
x=(10+- sqrt(100-4*7))/2
x=(10+- sqrt(72))/2
x=(10+- 6*sqrt(2))/2
x=5+- 3sqrt(2)
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for x=5+3sqrt(2), y=5-3sqrt(2)
-
for x=5-3sqrt(2), y=5+3sqrt(2)
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