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 2

Explanation:

Step1: Define variables and equations

Let the two numbers be \( x \) and \( y \). We know that \( xy = 42 \) and \( x + y=13 \). From the second equation, we can express \( y = 13 - x \) and substitute it into the first equation.
So we get \( x(13 - x)=42 \).

Step2: Solve the quadratic equation

Expand the left - hand side: \( 13x - x^{2}=42 \).
Rearrange it to the standard quadratic form: \( x^{2}-13x + 42 = 0 \).
Factor the quadratic equation. We need two numbers that multiply to \( 42 \) and add up to \( 13 \). We check the factor pairs of \( 42 \): \( 1\times42 = 42 \), \( 1 + 42=43
eq13 \); \( 2\times21 = 42 \), \( 2+21 = 23
eq13 \); \( 3\times14 = 42 \), \( 3 + 14=17
eq13 \); \( 6\times7 = 42 \), \( 6+7 = 13 \).
So the quadratic factors as \( (x - 6)(x - 7)=0 \).
Setting each factor equal to zero gives \( x-6 = 0 \) 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 \( 6 \)).