(13x+2)(5x-7)=(3x-4)

Solution for (13x+2)(5x-7)=(3x-4) equation:

(13x+2)(5x-7)=(3x-4)

We move all terms to the left:

(13x+2)(5x-7)-((3x-4))=0

We multiply parentheses ..

(+65x^2-91x+10x-14)-((3x-4))=0


We calculate terms in parentheses: -((3x-4)), so:

(3x-4)

We get rid of parentheses

3x-4

Back to the equation:

-(3x-4)


We get rid of parentheses

65x^2-91x+10x-3x-14+4=0

We add all the numbers together, and all the variables

65x^2-84x-10=0

a = 65; b = -84; c = -10;
Δ = b2-4ac
Δ = -842-4·65·(-10)
Δ = 9656
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:

$\sqrt{\Delta}=\sqrt{9656}=\sqrt{4*2414}=\sqrt{4}*\sqrt{2414}=2\sqrt{2414}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-84)-2\sqrt{2414}}{2*65}=\frac{84-2\sqrt{2414}}{130}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-84)+2\sqrt{2414}}{2*65}=\frac{84+2\sqrt{2414}}{130}
$