Question

writing and solving an equation to find negative integers
two negative integers are 5 units apart on the number line, and their product is 126. what is the sum of the two integers?
-23
-5
9
14

Explanation:

Step1: Define variables

Let the two negative integers be \( x \) and \( y \), where \( x < y \) (since they are negative, the larger one is closer to zero). The distance between them is 5, so \( y - x = 5 \) (because \( y > x \) and both are negative, \( y - x \) is positive 5). Their product is 126, so \( xy = 126 \). From \( y - x = 5 \), we can express \( y = x + 5 \).

Step2: Substitute into product equation

Substitute \( y = x + 5 \) into \( xy = 126 \):
\[
x(x + 5) = 126
\]
Expand the left side:
\[
x^2 + 5x - 126 = 0
\]

Step3: Solve the quadratic equation

We can factor the quadratic or use the quadratic formula. Let's try factoring. We need two numbers that multiply to -126 and add to 5. The numbers are 14 and -9 (since \( 14 \times (-9) = -126 \) and \( 14 + (-9) = 5 \)). So:
\[
(x + 14)(x - 9) = 0
\]
Wait, no, that's not right. Wait, the quadratic is \( x^2 + 5x - 126 = 0 \). Let's use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 5 \), \( c = -126 \).
\[
x = \frac{-5 \pm \sqrt{25 + 504}}{2} = \frac{-5 \pm \sqrt{529}}{2} = \frac{-5 \pm 23}{2}
\]
We have two solutions:
\[
x = \frac{-5 + 23}{2} = \frac{18}{2} = 9 \quad \text{(but we need negative integers, so discard)}
\]
\[
x = \frac{-5 - 23}{2} = \frac{-28}{2} = -14
\]

Step4: Find the other integer

Since \( y = x + 5 \), substitute \( x = -14 \):
\[
y = -14 + 5 = -9
\]

Step5: Find the sum

The two integers are -14 and -9. Their sum is \( -14 + (-9) = -23 \).

Answer:

-23