Question 1207404
<pre>
Find the real solutions, if any, of the given equation. 

(3/4)x^2 - (1/4)x - (1/2) = 0

Let me see.

I can use the quadratic formula. In the formula, a = 3/4, b = -1/4 and c = -1/2.

Yes? 

You could but it'll be a bit tedious, with all those fractions. It's much easier to get rid of the fractions by 
multiplying the equation by its LCD, 4. This'll give you: {{{matrix(2,3, (3/4)x^2 - (1/4)x - 1/2, "=", 0, 3x^2 - x - 2, "=", 0)}}}

Now, you can use the quadratic equation formula, but this quadratic (as is obvious, MAYBE), can be factored by
using the "ac" method, which requires 2 factors that have a PRODUCT of - 6 (a * c = + 3 * - 2), and a SUM of - 1
(b, or the coefficient on "x"). These 2 FACTORS are - 3 and + 2. 
We then get the following: 3x<sup>2</sup> - x - 2 = 0 
                     3x<sup>2</sup> - 3x + 2x - 2 = 0 ----- Replacing - x with - 3x + 2x
                     <u>3x<sup>2</sup> - 3x</u> <u>+ 2x - 2</u> = 0 ----- PAIRing binomials
                  3x(x - 1) + 2(x - 1) = 0
                       (3x + 2)(x - 1) = 0 ----- FACTORED form
              3x + 2 = 0       | x - 1 = 0
                  3x = - 2     |     x = 0 + 1
                  {{{highlight_green(matrix(1,3, x, "=", highlight(- 2/3)))}}}   OR   {{{highlight_green(matrix(1,3, x, "=", highlight(1)))}}}</pre>