Question

write two numbers that multiply to the value on top and add to the value on bottom
42
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-13
answer attempt 1 out of 10
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Explanation:

Step1: Define variables and equations

Let the two numbers be \( x \) and \( y \). We have two equations:
\( xy = 42 \) (they multiply to 42) and \( x + y = 13 \) (they add to 13).
From the second equation, we can express \( y = 13 - x \).

Step2: Substitute and solve the quadratic equation

Substitute \( y = 13 - x \) into the first equation:
\( x(13 - x)=42 \)
Expand: \( 13x - x^{2}=42 \)
Rearrange into standard quadratic form: \( x^{2}-13x + 42 = 0 \)
Factor the quadratic: We need two numbers that multiply to 42 and add to - 13 (for the quadratic \( x^{2}-13x + 42 \)). The factors of 42 are 1 & 42, 2 & 21, 3 & 14, 6 & 7. Among these, 6 and 7 multiply to 42 and add to 13. So the factored form is \( (x - 6)(x - 7)=0 \)
Set each factor equal to zero: \( x - 6 = 0 \) or \( x - 7 = 0 \)
So \( x = 6 \) or \( x = 7 \)
If \( x = 6 \), then \( y=13 - 6 = 7 \)
If \( x = 7 \), then \( y = 13 - 7=6 \)

Answer:

The two numbers are 6 and 7 (or 7 and 6). So the boxes can be filled with 6 and 7 (in either order).