Question

write two numbers that multiply to the value on top and add to the value on bottom. -87 × + -26

Explanation:

Step1: Define variables

Let the two numbers be \( x \) and \( y \). We know that \( xy=-87 \) and \( x + y=-26 \).

Step2: Rewrite the linear equation

From \( x + y=-26 \), we can express \( y=-26 - x \).

Step3: Substitute into the product equation

Substitute \( y=-26 - x \) into \( xy=-87 \):
\( x(-26 - x)=-87 \)
\( -26x-x^{2}=-87 \)
\( x^{2}+26x - 87 = 0 \)

Step4: Factor the quadratic equation

We need to find two numbers that multiply to \( - 87\) and add to \(26\). The factors of \(87\) are \(1, 3, 29, 87\). We can rewrite the quadratic as \(x^{2}+29x - 3x-87 = 0\)
\(x(x + 29)-3(x + 29)=0\)
\((x + 29)(x - 3)=0\)

Step5: Solve for x

Setting each factor equal to zero:
\(x+29 = 0\) gives \(x=-29\)
\(x - 3=0\) gives \(x = 3\)

Step6: Find y

If \(x=-29\), then \(y=-26-(-29)=3\)
If \(x = 3\), then \(y=-26 - 3=-29\)

Answer:

The two numbers are \(-29\) and \(3\) (or \(3\) and \(-29\))