7(t+5t+9)+t=t(7t-2)13

Solution for 7(t+5t+9)+t=t(7t-2)13 equation:

7(t+5t+9)+t=t(7t-2)13

We move all terms to the left:

7(t+5t+9)+t-(t(7t-2)13)=0

We add all the numbers together, and all the variables

7(6t+9)+t-(t(7t-2)13)=0

We add all the numbers together, and all the variables

t+7(6t+9)-(t(7t-2)13)=0

We multiply parentheses

t+42t-(t(7t-2)13)+63=0


We calculate terms in parentheses: -(t(7t-2)13), so:

t(7t-2)13

We multiply parentheses

91t^2-26t

Back to the equation:

-(91t^2-26t)


We add all the numbers together, and all the variables

43t-(91t^2-26t)+63=0

We get rid of parentheses

-91t^2+43t+26t+63=0

We add all the numbers together, and all the variables

-91t^2+69t+63=0

a = -91; b = 69; c = +63;
Δ = b2-4ac
Δ = 692-4·(-91)·63
Δ = 27693
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{27693}=\sqrt{9*3077}=\sqrt{9}*\sqrt{3077}=3\sqrt{3077}$
$t_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(69)-3\sqrt{3077}}{2*-91}=\frac{-69-3\sqrt{3077}}{-182}
$
$t_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(69)+3\sqrt{3077}}{2*-91}=\frac{-69+3\sqrt{3077}}{-182}
$