Hi, there---
Your Problem:
Four times the square of a number is 21 more than eight times the number. What is the number?
A Solution:
STEP 1. Our first task is to turn these words into algebraic expressions and equations. We are asked to
find an unknown number. Let's assign that number a variable. I'll call it n, but you could choose any letter.
"The square of the number" is .
"Four times the square of the number" is .
"Eight times the number" is .
"21 more than eight times the number is .
STEP 2. Now we make an equation to show that "four times the square of the number" and "21 more than
eight times the number" have the same value. In other words
STEP 3. We have a quadratic equation. We will set it equal to zero. Then we will solve it by factoring, or
by using the quadratic formula.
Subtract 8n from both sides of the equation. Then subtract 21 from both sides of the equation.
Now we have a quadratic equation equal to zero. Let's see if we can factor it. The factors of 4 are 4 and
1 or 2 and 2. The factors of 21 are 3 and 7 or 21 and 1. We play around with these to find a sum of -8.
We find that the equation in factored form is
Therefore, or .
Our solutions are or .
STEP 4. We need to check out solutions in the original problem. Our first solution is n = -3/2. Substitute
-3/2 for "a number" in the words of the problem.
Four times the square of -3/2 is (4)(9/4) which equals 9.
21 more than 8 times -3/2 is (8)(-3/2) + 21) which is 9.
Now, we check our second solution:
Four times the square of 7/2 is (4)(49/4) which equals 49.
21 more than 8 times 7/2 is (8)(7/2) + 21) which is 49.
Therefore, two different numbers make this puzzle true. The number is either -3/2 or 7/2.
Hope this helps. Feel free to email me if you have a question.
Mrs. F
math.in.the.vortex@gmail.com
