You've probably known how to find an average for a long time. You add the numbers up and then divide by the number of numbers. You will make a big leap forward in your ability to solve word problems when you learn that the logic of what you do does not change, regardless of what the numbers look like!. So to find the average of our two numbers, you add them:
and then divide by 2 (since there are two numbers):
So the expression for the average is .
The last number in the problem is "5 times the number" which would be 5*x.
Now that we have an expression for each of the numbers, we need an equation. Sometimes we get equations from formulas. Sometimes we get equations from translating a sentence in the problem. In this problem we will translate the sentence: "The average of a non-zero number and its square is 5 times the number."
"The average of a non-zero number and its square" translates into:
"is" (and other forms of the verb "to be") translates into:
=
and "5 times the number" translates into
5x
This makes the full translation of "The average of a non-zero number and its square is 5 times the number.":
Look at this and try to understand how the equation is a translation of the sentence.
Now that we have an equation we can solve the problem. We'll simplify things first by getting rid of the fraction (by multiplying each side by 2):
Now, since this is a quadratic equation (because of the squared term), we want one side to be zero. So we'll subtract 10x from each side:
Now we factor (or use the Quadratic Formula). This factors easily:
x*(-9 + x) = 0
From the Zero Product Property we know that this (or any) product can be zero only if one (or more) of the factors is zero. So:
x = 0 or -9 + x = 0
Solving the second equation we get:
x = 0 or x = 9
Remembering "a non-zero number" we must reject the first solution because it says that x is zero. So the only solution for this problem is 9.
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