Question 648116
Original problem: {{{4x^2-4x-99}}}

We're going to be factoring this to extract the roots (what 'x' equals)

Step 1: We must find two numbers that when '4' and '-99' multiply, will divide to give you '-4' in the middle.

{{{4x^2-4x-99}}}

You have quite a few number of combinations:

The {{{4x^2}}} can be broken up into

{{{(4x  )(x  )}}}
or
{{{(2x  )(2x  )}}}

We know that '-99' has six factors, and these include:

1 x 99, 3 x 33, and 9 x 11

Step 2: Let's start trying to see what will work and not work (trial and error method)

We know that regardless of signs, none of the factors of '-99' will be able to multiply here to give you a '-4' in the middle using this equation:

{{{(4x + 3 )(x - 33 )}}} ---> Does not work, gives you {{{-33x + 3x}}} in the middle
{{{(4x - 3 )(x + 33 )}}} ---> Does not work, gives you {{{33x - 3x}}} in the middle
{{{(4x +  1 )(x - 99 )}}} ---> Does not work, gives you {{{-396x + x}}} in the middle
{{{(4x - 1 )(x + 99 )}}} ---> Does not work, gives you {{{396x - x}}} in the middle
{{{(4x + 9 )(x - 11 )}}} ---> Does not work, gives you {{{-44x + 9x}}} in the middle
{{{(4x - 9 )(x + 11 )}}} ---> Does not work, gives you {{{44x - 9x}}} in the middle

Now that we know that the {{{(4x  )(x  )}}} equation won't work, let's move on to the other equation that could possibly work..

Step 3: Let's start trying to see what will work and not work (trial and error method)

We know that '-99' has six factors, and these include:

1 x 99, 3 x 33, and 9 x 11

We know that regardless of signs, none of the factors of '-99' will be able to multiply here to give you a '-4' in the middle using this equation:

{{{(2x + 1 )(2x - 99 )}}}  ---> Does not work, gives you {{{-198x + 2x}}} in the middle
{{{(2x - 1 )(2x + 99 )}}} ---> Does not work, gives you {{{198x - 2x}}} in the middle
{{{(2x + 3 )(2x - 33 )}}} ---> Does not work, gives you {{{-66x + 6x}}} in the middle
{{{(2x - 3 )(2x + 33 )}}} ---> Does not work, gives you {{{66x - 6x}}} in the middle
{{{(2x + 9 )(2x - 11 )}}} --->  Works!!  Gives you {{{-22x + 18x}}} in the middle, which is equivalent to '-4x'
{{{(2x - 9 )(2x + 11 )}}} ---> Does not work, gives you {{{22x - 18x}}} in the middle

So.. now we know that the factored version of the equation you provided is:

{{{(2x + 9 )(2x - 11 )}}}

To get the roots, you must set each parenthesis equal to zero like so..

Step 1: Set each parenthesis equal to zero

{{{(2x + 9 )=0}}}
{{{(2x - 11 )=0}}}

Step 2: Let's solve each one at a time

{{{(2x + 9 )=0}}}

Subtract '9' from both sides to get the '2x' by itself

{{{(2x + 9 )=0}}}
{{{2x=-9}}}

Divide both sides by '2' to get 'x' by itself

{{{2x=-9}}}
{{{x=(-9/2)}}}

First root is {{{x=(-9/2)}}}

Step 3: Let's find the other root

{{{(2x - 11 )=0}}}

Add '11' to both sides to get the '2x' by itself

{{{(2x - 11 )=0}}}
{{{2x=11}}}

Divide both sides by '2' to get the 'x' by itself

{{{2x=11}}}
{{{x=(11/2)}}}

Second root is {{{x=(11/2)}}}

So, the factored version of this equation is {{{(2x + 9 )(2x - 11 )}}} and its roots are {{{x=(-9/2)}}} and {{{x=(11/2)}}}