Question

What are the solutionso of 8x^2 = 6 + 22x

Answer 1

8x² = 6 + 22x

Rearranging the quadratic equation, (Ax² + Bx + C = 0)
                                            8x² - 22x - 6 = 0

Simplify. Divide both sides of the equation by 2.
                                           4x² - 11x - 3 = 0

Using the quadratic formula. Identify first the value of the coefficients. 
A = 4
B = -11
C = -3

Using the quadratic formula:

$x = \frac{-B +- \sqrt{B^2 - 4AC} }{2A}$

Substitute A, B and C

$x_{1} = \frac{-(-11) + \sqrt{(-11)^2 - 4(4)(-3)} }{2(4)}$
     
                 x = 3

$x_{1} = \frac{-(-11) - \sqrt{(-11)^2 - 4(4)(-3)} }{2(4)}$

                 x = $- \frac{1}{4}$



                    

Answer 2

You can get all the terms on one side, and then solve for x by any means:
$0 = -8x^2+22x+6$

Factor out a -2:
$0 = -2(4x^2-11x-3)$

Factor this equation:
$0 = 4x^2-11x-3$

I will use the AC method. To use it, first multiply a and c (in ax^2 + bx +c):
$4*-3 = -12$

Now, look for two numbers that multiply to -12 and add to -11. Obviously these numbers are -12 and 1. (-12*1 = -12 and -12+1  = -11). Now, because of rules, you set it up in a Punnett square:

4x^2        -12x
x              -3

Now, we find common factors of the terms in rows:
     _x_______-3__
4x| 4x^2        -12x
1   | x              -3

So, you can use this to write an equivalent expression to the quadratic given:
$(x-3)(4x+1)$

Now, we known the factors are (solve for x): 3 and -1/4.