53=3/4x^2+4=4x^2

Solution for 53=3/4x^2+4=4x^2 equation:

53=3/4x^2+4=4x^2

We move all terms to the left:

53-(3/4x^2+4)=0


Domain of the equation: 4x^2+4)!=0

x∈R


We get rid of parentheses

-3/4x^2-4+53=0

We multiply all the terms by the denominator


-4*4x^2+53*4x^2-3=0

Wy multiply elements

-16x^2+212x^2-3=0

We add all the numbers together, and all the variables

196x^2-3=0

a = 196; b = 0; c = -3;
Δ = b2-4ac
Δ = 02-4·196·(-3)
Δ = 2352
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:

$\sqrt{\Delta}=\sqrt{2352}=\sqrt{784*3}=\sqrt{784}*\sqrt{3}=28\sqrt{3}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-28\sqrt{3}}{2*196}=\frac{0-28\sqrt{3}}{392}
=-\frac{28\sqrt{3}}{392}
=-\frac{\sqrt{3}}{14}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+28\sqrt{3}}{2*196}=\frac{0+28\sqrt{3}}{392}
=\frac{28\sqrt{3}}{392}
=\frac{\sqrt{3}}{14}
$