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

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

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

We move all terms to the left:

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

We add all the numbers together, and all the variables

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

We get rid of parentheses

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

We multiply all the terms by the denominator


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

We add all the numbers together, and all the variables

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

Wy multiply elements

-2x^2+14x+9=0

a = -2; b = 14; c = +9;
Δ = b2-4ac
Δ = 142-4·(-2)·9
Δ = 268
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:

$\sqrt{\Delta}=\sqrt{268}=\sqrt{4*67}=\sqrt{4}*\sqrt{67}=2\sqrt{67}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(14)-2\sqrt{67}}{2*-2}=\frac{-14-2\sqrt{67}}{-4}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(14)+2\sqrt{67}}{2*-2}=\frac{-14+2\sqrt{67}}{-4}
$