Find two numbers that have a sum of 22 and a product of 112

Let the two numbers be \( x \) and \( y \). We are given the following conditions:

1. The sum of the numbers:
   \[
   x + y = 22
   \]
2. The product of the numbers:
   \[
   x \cdot y = 112
   \]

We can express \( y \) in terms of \( x \) using the first equation:
\[
y = 22 - x
\]

Substitute this expression for \( y \) into the second equation:
\[
x \cdot (22 - x) = 112
\]

Now, expand and rearrange the equation:
\[
x(22 - x) = 112 \quad \Rightarrow \quad 22x - x^2 = 112
\]
\[
-x^2 + 22x - 112 = 0
\]

Multiply the entire equation by \(-1\) to make it a standard quadratic form:
\[
x^2 - 22x + 112 = 0
\]

Now, solve this quadratic equation using the quadratic formula:
\[
x = \frac{-(-22) \pm \sqrt{(-22)^2 - 4(1)(112)}}{2(1)}
\]
\[
x = \frac{22 \pm \sqrt{484 - 448}}{2}
\]
\[
x = \frac{22 \pm \sqrt{36}}{2}
\]
\[
x = \frac{22 \pm 6}{2}
\]

So, the two possible solutions for \( x \) are:
\[
x = \frac{22 + 6}{2} = 14 \quad \text{or} \quad x = \frac{22 - 6}{2} = 8
\]

Thus, the two numbers are 14 and 8.

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Find two numbers that have a sum of 22 and a product of 112.

Discover how to solve a quadratic equation to find two numbers with a sum of 22 and a product of 112. This step-by-step solution covers important algebraic concepts like quadratic equations and the quadratic formula. Ideal for students in Grades 8-10.

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