Question

Four times the square of a certain number increased by 6 times the number equals 108. What is the number?

Answer 1

`x=9/2`
`x=-6`

Explanation:

Let the number be `x`

Four times the square of a certain number is `4*x^2=4x^2`

Increased (addition) by `6` times the number is `+6*x=+6x`

Equals `108` is `=108`

Let's put it all together!

`4x^2+6x=108`

Solve for `x`

First, make sure the equation is in the standard form

a`x^2` + bx + c = 0

Our equation is not in standard form, let's rewrite that.

`4x^2+6x-108=0`

If `a!= 1`, the `ac` method is applied.

In this case, `a= 4` so we have to use the `ac` method.

What is the ac method???

The ac method is simply the value of a times the value of c

`a= 4`

`b= 6`

`c=-108`

`a*c= -432`

Note that computing the value would be a headache. I would advise that you simplify the equation by dividng each term by the lcm which in our case happens to be `2`

So the equation we will work with is;

`2x^2+3x-54=0`

Let's find `ac` now

`a=2`

`b=3`

`c=-54`

`a*c=-108`

Now we have to find the numbers that will give a product of -108 as well as a sum of 3.

`12*-9=-108 (ac)`
`12+(-9)= 3 (b)`

Bingo! Now that we have these numbers, we rewrite the equation substituting `12` and `-9` for b.

2`x^2` + 12x - 9x - 54 = 0 ...this is the same as the inital equation when simplified.

Now, we factor by grouping and pulling out the gcf .

`(2x^2 +12x) + (-9x- 54) = 0`

`2x(x+6) -9(x+6) = 0`

Note that `-9` factored out leaves `x+6` because

`-9*+x=-9x` and `-9*+6=-54`

So our factors are

`(x+6)(2x-9) = 0`

Solve for x
`2x-9=0`
`2x=9`

Dividing both sides by 2, we get
`(cancel(2)x)/cancel2 = (9)/2`
`x=9/2`

Solve the other also for x
`x+6 = 0`
`x=-6`