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:
Identify the a, b and c in the equation x2-4x-12=0.
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.
Find ax^2 times c. ax^2 is x^2 for our problem and c is -12.
x^2 times -12 is -12x2.
Next we find the factors of ax^2 times c which when added give us the middle term (bx which is -4x).
Numbers that multiply to -12x2 are -x & 12x, x & -12x, 2x & -6x, -2x & 6x, 3x & -4x, -3x & 4x.
Notice that the factors 2x & -6x add up to out middle term (-4x).
We replace the middle term with the values we found in the previous step.
x2+2x-6x-12=0
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).
x^2 and 2x are both divisible by x.
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.
x(x+2)-6x-12=0
We then factor the next two terms. We find the common factor just like the previous step.
The common factor of -6x and -12 is -6
We take out -6 and in a set of parentheses write what we get after dividing -6x and -12x by -6.
x(x+2)-6(x+2)=0
We then factor out what is common between the two groups.
x(x+2) and -6(x+2) have x+2 as the common factor.
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.
(x+2)(x-6)=0
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.
So if x+2 times x-6 is 0, either x+2 is 0 or x-6 is 0.
x+2=0 so x=-2
x-6=0 so x=6
x=6 or x=-2 But the problem only wants the positive value so x will only be 6
Step-by-step explanation:
