a(2a-9)=5(3a-7)-1

Solution for a(2a-9)=5(3a-7)-1 equation:

a(2a-9)=5(3a-7)-1

We move all terms to the left:

a(2a-9)-(5(3a-7)-1)=0

We multiply parentheses

2a^2-9a-(5(3a-7)-1)=0


We calculate terms in parentheses: -(5(3a-7)-1), so:

5(3a-7)-1

We multiply parentheses

15a-35-1

We add all the numbers together, and all the variables

15a-36

Back to the equation:

-(15a-36)


We get rid of parentheses

2a^2-9a-15a+36=0

We add all the numbers together, and all the variables

2a^2-24a+36=0

a = 2; b = -24; c = +36;
Δ = b2-4ac
Δ = -242-4·2·36
Δ = 288
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{288}=\sqrt{144*2}=\sqrt{144}*\sqrt{2}=12\sqrt{2}$
$a_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-24)-12\sqrt{2}}{2*2}=\frac{24-12\sqrt{2}}{4}
$
$a_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-24)+12\sqrt{2}}{2*2}=\frac{24+12\sqrt{2}}{4}
$