Question

How do you solve `6x² - 4 = 5x`?

Answer 1

`x=-1/2, 4/3`

Explanation:

`6x^2-4=5x`

Gather all terms on the left side.

`6x^2-5x-4=0`

This is a quadratic equation, `ax^2+bx+c`, where `a=6, b=-5, and c=-4`.

You can use the quadratic formula to solve this equation.

`x=(-b+-sqrt(b^2-4ac))/(2a)`

`x=(-(-5)+-sqrt(-5^2-4*6*-4))/(2*6)` =

`x=(5+-sqrt(25+96))/12` =

`x=(5+-sqrt(121))/12` =

`x=(5+-11)/12`

Solve for `x`.

`x=(5+11)/12=16/12=4/3`

`x=(5-11)/12=-6/12=-1/2`

`x=-1/2, 4/3`

Answer 2

Solve y = 6x^2 - 5x - 4 = 0 (1)
Ans: - 1/2 and 4/3

Explanation:

I use the new Transforming Method.
Transformed y' = x^2 - 5x - 24. (2). Roots have opposite signs.
Factor pairs of (-24) -> (-2, 12)(-3, 8). This sum is 5 = -b. Two real roots of (2) are: y1 = -3 and y2 = 8.
Back to original equation (1), the 2 real roots are:
`x1 = y1/a = - 3/6 = -1/2` and `x2 = y2/a = 8/6 = 4/3`