The difference of the square of a number and 12 is equal to 4 times that number. Find the positive solution.

Answer 1

Question: The difference of the square of a number and 12 is equal to 4 times that number. Find the positive solution.

Answer:

Creating the Equation:

Step 1: Consider the unknown number to be x.

Step 2: Find out the first operation being done to x.

The problem states, "square of a number" which means the first operation we should apply to x is square it: x^2.

Step 3: Find the next operation being done to x.

The problem states, "The difference of the square of a number and 12". The word "difference" suggests subtraction: x^2-12.

Step 4: Next we look at what x is equal.

The problem mentions, "equal to 4 times that number". We consider "that number" or the unknown number to be x. So we get 4x.

Step 5: Formulate the equation.

For, "the difference of the square of a number and 12" we got the expression x^2-12 and for "4 times that number", we got 4x. The problem says these are equal to one another. So our equation is x^2-12=4x.

Solving the Equation:

Step 6: To solve, we will first make one side of the equation 0. In order to do that, we subtract 4x from both sides of the equation.

x^2 -12 = 4x

-4x -4x

-----------------

x^2-4x-12=0

Step 7: Now that we have our quadratic in standard form (ax^2+bx+c=0), we can solve using the quadratic formula or we can factor.

a) Using the Quadratic Formula:

The quadratic formula is: -b±√b²-4ac

x= ——————

2a

In order to use it, we first need to identify the a, b and c in our equation.

We have x²-4x-12. Here a (the number before x^2) is 1, b (the number before x) is -4 and c (the constant or number without any variables) is -12.

Now we plug it into the formula and simplify to find x.

-(-4)±√(-4)²-4(1)(-12) 4±√16-4(1)(-12) 4±√16+48 4±√64 4±8

x=—————————— = ———————— = —————— = —————— = ————

2(1) 2 2 2 2

4±8 4+8 4-8

—— means x= —— or x= ——

2 2 2

After simplifying we get: 12 -4

x= ——— = 6 or x= ——— = -2

2 2

x= 6 or -2

Since the problem asks for the positive value of x, it will only be 6.

b) Solving by Factoring:

1. Identify the a, b and c in the equation x²-4x-12=0.
2. Here a (the number before x^2) is 1, b (the number before x) is -4 and c (the constant or number without any variables) is -12.
3. Find ax^2 times c. ax^2 is x^2 for our problem and c is -12.
4. x^2 times -12 is -12x².
5. Next we find the factors of ax^2 times c which when added give us the middle term (bx which is -4x).
6. Numbers that multiply to -12x² are -x & 12x, x & -12x, 2x & -6x, -2x & 6x, 3x & -4x, -3x & 4x.
7. Notice that the factors 2x & -6x add up to out middle term (-4x).
8. We replace the middle term with the values we found in the previous step.
9. x²+2x-6x-12=0
10. We factor the first two terms. In order to do that, we find at the common factor (number or variable that can divide both of the first terms).
11. x^2 and 2x are both divisible by x.
12. So we can take out x as the common factor and inside a set of parentheses write what is remaining after we divide x² and 2x by x.
13. x(x+2)-6x-12=0
14. We then factor the next two terms. We find the common factor just like the previous step.
15. The common factor of -6x and -12 is -6
16. We take out -6 and in a set of parentheses write what we get after dividing -6x and -12x by -6.
17. x(x+2)-6(x+2)=0
18. We then factor out what is common between the two groups.
19. x(x+2) and -6(x+2) have x+2 as the common factor.
20. We take out x+2 as the common factor and write what we get after dividing x(x+2) and -6(x+2) by x+2 in another set of parentheses.
21. (x+2)(x-6)=0
22. Use the zero product property to solve. Zero product property states if a times b equals 0, then a is 0 or b is 0.
23. So if x+2 times x-6 is 0, either x+2 is 0 or x-6 is 0.
24. x+2=0 so x=-2
25. x-6=0 so x=6
26. x=6 or x=-2 But the problem only wants the positive value so x will only be 6.

Answer 2

Word problems can be tricky! The hardest part of a word problem is to successfully translate it into a recognizable numeric expression. In this case, we see that:

1. "the square of a number" can be notated as "x²"
2. "four times that number" can be notated as "4x"

Put together we see that "the difference of the square of a number and 12" can be written as "x² - 12". Then the second clause "is equal to 4 times that number" gives us the whole algebraic equation: x² - 12 = 4x.

Solving the quadratic equation we recall step by step looks like this:

x² - 12 = 4x

x² - 4x - 12 = 0 /subtracting 4x from both sides

(x - 6)(x + 2) = 0 /factoring using the quadratic equation

The question asks for the positive solution, which we find by solving for (x - 6) = 0. We find x=6.