Question

Find the roots of equation, if they exists, by applying the quadratic formula :

$12abx {}^{2} - (9a {}^{2} - 8b {}^{2} )x - 6ab = 0$

, a # 0 and b # 0

Answer 1

hey mate
here's the solution

Answer 2

12abx^2  - (9a^2 – 8b^2)x – 6ab = 0. 

Roots of the quadratic equatios ax^2 + bx + c = 0 are =[-b ± (b^2 - 4ac)] / 2a 

From the given equation, a = 12ab, b = - (9a^2 – 8b^2), c = - 6ab 

Roots = (9a^2 – 8b^2) ± √[(9a^2 – 8b^2)^2 + 4(12ab)(6ab)] / 2(12ab) 
= (9a^2 – 8b^2) ± √(81a^4 + 64 b^4 - 144 a^2b^2) + (288a^2b^2)] / 24ab 
= (9a^2 – 8b^2) ± √[(81a^4 + 64 b^4 + 144 a^2b^2)] / 24ab = (9a^2 – 8b^2) ± √[(9a^2 + 8b^2)^2] / 24ab
= (9a^2 – 8b^2) ± (9a^2 + 8b^2) / 24ab 
= (9a^2 – 8b^2) + (9a^2 + 8b^2) / 24ab, (9a^2 – 8b^2) - (9a^2 + 8b^2) / 24ab 
= 18a^2/ 24ab, – 16b^2 / 24ab 
= 3a/4b, - 2b/3a 

Hence, the roots of the quadratic equation are 3a/4b, - 2b/3a.