Question

The product of two consecutive integers is 72. Using x to represent the smaller integer, translate the problem into a variable expression and solve for x.

Answer 1

Let's denote the smaller integer as \( x \). The next consecutive integer would then be \( x + 1 \).

The product of these two consecutive integers can be expressed as:

\[
x(x + 1) = 72
\]

Now we need to solve this equation. Expanding the left side gives:

\[
x^2 + x = 72
\]

Next, we rearrange the equation to set it to zero:

\[
x^2 + x - 72 = 0
\]

Now, we can factor the quadratic equation. We need to find two numbers that multiply to \(-72\) and add to \(1\). The numbers \( 9 \) and \(-8\) work because:

\[
9 \times (-8) = -72 \quad \text{and} \quad 9 + (-8) = 1
\]

Thus, we can factor the equation as:

\[
(x + 9)(x - 8) = 0
\]

Setting each factor equal to zero gives us the possible solutions for \( x \):

1. \( x + 9 = 0 \) which implies \( x = -9 \)
2. \( x - 8 = 0 \) which implies \( x = 8 \)

Since \( x \) represents a smaller integer, we have two solutions for \( x \): \( -9 \) and \( 8 \).

If we check:

- For \( x = -9 \), the consecutive integers are \(-9\) and \(-8\):
\[
-9 \times -8 = 72
\]

- For \( x = 8\), the consecutive integers are \(8\) and \(9\):
\[
8 \times 9 = 72
\]

Both pairs of integers correctly yield a product of 72.

Thus, the values of \( x \) are:

\[
\boxed{-9} \quad \text{and} \quad \boxed{8}
\]