10x+1=5x^2

Solution for 10x+1=5x^2 equation:

10x+1=5x^2

We move all terms to the left:

10x+1-(5x^2)=0

determiningTheFunctionDomain
-5x^2+10x+1=0

a = -5; b = 10; c = +1;
Δ = b2-4ac
Δ = 102-4·(-5)·1
Δ = 120
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:

$\sqrt{\Delta}=\sqrt{120}=\sqrt{4*30}=\sqrt{4}*\sqrt{30}=2\sqrt{30}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(10)-2\sqrt{30}}{2*-5}=\frac{-10-2\sqrt{30}}{-10}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{30}}{2*-5}=\frac{-10+2\sqrt{30}}{-10}
$