Question

write two numbers that multiply to the value on top and add to the value on bottom.
-70
×
□
□
+
3

Explanation:

Step1: Define variables and equations

Let the two numbers be \( x \) and \( y \). We have two equations:

1. \( x \times y = -70 \)
2. \( x + y = 3 \)

From the second equation, we can express \( y = 3 - x \) and substitute it into the first equation.

Step2: Substitute and solve the quadratic equation

Substitute \( y = 3 - x \) into \( x \times y = -70 \):
\[
x(3 - x) = -70
\]
Expand the left side:
\[
3x - x^2 = -70
\]
Rearrange to form a quadratic equation:
\[
x^2 - 3x - 70 = 0
\]
Factor the quadratic equation. We need two numbers that multiply to \( -70 \) and add to \( -3 \). The numbers are \( -10 \) and \( 7 \) (since \( -10 \times 7 = -70 \) and \( -10 + 7 = -3 \), but we need to adjust for the correct signs). Wait, actually, we can factor \( x^2 - 3x - 70 \) as \( (x - 10)(x + 7) = 0 \)? Wait, no, let's check: \( (x - 10)(x + 7) = x^2 + 7x - 10x - 70 = x^2 - 3x - 70 \), yes. So the solutions are \( x = 10 \) or \( x = -7 \).

If \( x = 10 \), then \( y = 3 - 10 = -7 \). Let's check: \( 10 \times (-7) = -70 \) and \( 10 + (-7) = 3 \). Correct.

If \( x = -7 \), then \( y = 3 - (-7) = 10 \). Which is the same pair of numbers, just reversed.

Answer:

The two numbers are \( 10 \) and \( -7 \) (or \( -7 \) and \( 10 \)). So the boxes can be filled with \( 10 \) and \( -7 \) (or \( -7 \) and \( 10 \)).