(1/4)=2x^2+9x-7

Solution for (1/4)=2x^2+9x-7 equation:

(1/4)=2x^2+9x-7

We move all terms to the left:

(1/4)-(2x^2+9x-7)=0

We add all the numbers together, and all the variables

-(2x^2+9x-7)+(+1/4)=0

We get rid of parentheses

-2x^2-9x+7+1/4=0

We multiply all the terms by the denominator


-2x^2*4-9x*4+1+7*4=0

We add all the numbers together, and all the variables

-2x^2*4-9x*4+29=0

Wy multiply elements

-8x^2-36x+29=0

a = -8; b = -36; c = +29;
Δ = b2-4ac
Δ = -362-4·(-8)·29
Δ = 2224
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{2224}=\sqrt{16*139}=\sqrt{16}*\sqrt{139}=4\sqrt{139}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-36)-4\sqrt{139}}{2*-8}=\frac{36-4\sqrt{139}}{-16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-36)+4\sqrt{139}}{2*-8}=\frac{36+4\sqrt{139}}{-16}
$