2x+3/(4x+16)=7

Solution for 2x+3/(4x+16)=7 equation:

2x+3/(4x+16)=7

We move all terms to the left:

2x+3/(4x+16)-(7)=0


Domain of the equation: (4x+16)!=0

We move all terms containing x to the left, all other terms to the right

4x!=-16

x!=-16/4

x!=-4

x∈R


We multiply all the terms by the denominator


2x*(4x+16)-7*(4x+16)+3=0

We multiply parentheses

8x^2+32x-28x-112+3=0

We add all the numbers together, and all the variables

8x^2+4x-109=0

a = 8; b = 4; c = -109;
Δ = b2-4ac
Δ = 42-4·8·(-109)
Δ = 3504
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:

$\sqrt{\Delta}=\sqrt{3504}=\sqrt{16*219}=\sqrt{16}*\sqrt{219}=4\sqrt{219}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-4\sqrt{219}}{2*8}=\frac{-4-4\sqrt{219}}{16}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+4\sqrt{219}}{2*8}=\frac{-4+4\sqrt{219}}{16}
$