Question

write two numbers that multiply to the value on top and add to the value on bottom.
-40
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6
answer attempt 1 out of 2
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Explanation:

Step1: Define variables and equations

Let the two numbers be \( x \) and \( y \). We have two equations:
\( xy = -40 \) (product is -40) and \( x + y = 6 \) (sum is 6).
From the second equation, we can express \( y = 6 - x \).

Step2: Substitute and solve the quadratic equation

Substitute \( y = 6 - x \) into the first equation:
\( x(6 - x) = -40 \)
Expand: \( 6x - x^2 = -40 \)
Rearrange: \( x^2 - 6x - 40 = 0 \)
Factor the quadratic equation: We need two numbers that multiply to -40 and add to -6. Wait, actually, for \( x^2 - 6x - 40 = 0 \), we factor as \( (x - 10)(x + 4) = 0 \) (because \( -10 \times 4 = -40 \) and \( -10 + 4 = -6 \), but wait, our original sum is 6. Wait, maybe I made a sign error. Let's redo the substitution.
Wait, original equations: \( xy = -40 \) and \( x + y = 6 \). So \( y = 6 - x \), so \( x(6 - x) = -40 \) => \( 6x - x^2 = -40 \) => \( x^2 - 6x - 40 = 0 \). Wait, using quadratic formula: \( x=\frac{6\pm\sqrt{36 + 160}}{2}=\frac{6\pm\sqrt{196}}{2}=\frac{6\pm14}{2}\). So \( x=\frac{6 + 14}{2}=10 \) or \( x=\frac{6 - 14}{2}=-4 \). So if \( x = 10 \), then \( y = 6 - 10 = -4 \). If \( x = -4 \), then \( y = 6 - (-4)=10 \). So the two numbers are 10 and -4.

Answer:

10 and -4 (or -4 and 10)