Name: Solving Free Fall Problem Using Quadratic Equation
Uploaded: 2019-07-19
Duration: 7 min 7 s
Description: In this video, we will learn how we can solve the problem of free fall effectively using the quadratic equation. Let's watch the video and learn.

Question

Find the roots of the equation:

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

Accepted Answer

Given equation: $12 a b x^{2} - (9 a^{2} - 8 b^{2}) x - 6 a b = 0$

Comparing it with $a x^{2} + b x + c = 0$ , we get,

$a = 12 a b, b = - (9 a^{2} - 8 b^{2}), c = - 6 a b$

We know that the discriminant of a quadratic equation is given by,

$D = b^{2} - 4 a c$

$D = {[- (9 a^{2} - 8 b^{2})]}^{2} - 4 (12 a b) (- 6 a b)$

$D = {(8 b^{2} - 9 a^{2})}^{2} + 288 a^{2} b^{2}$

$= (64 b^{4} + 81 a^{4} - 144 a^{2} b^{2}) + 288 a^{2} b^{2}$

$= 81 a^{4} + 64 b^{4} + 144 a^{2} b^{2}$

$D = {(9 a^{2} + 8 b^{2})}^{2}$

So, the discriminant of the quadratic equation is $D = {(9 a^{2} + 8 b^{2})}^{2}$

The roots of the quadratic equation is given by:

$x = \frac{- b \pm \sqrt{D}}{2 a}$

$x = \frac{- [- (9 a^{2} - 8 b^{2})] \pm \sqrt{{(9 a^{2} + 8 b^{2})}^{2}}}{2 (12 a b)}$

$x = \frac{(9 a^{2} - 8 b^{2}) \pm (9 a^{2} + 8 b^{2})}{24 a b}$

$x = \frac{(9 a^{2} - 8 b^{2}) + (9 a^{2} + 8 b^{2})}{24 a b}, \frac{(9 a^{2} - 8 b^{2}) - (9 a^{2} + 8 b^{2})}{24 a b}$

$x = \frac{18 a^{2}}{24 a b}, \frac{- 16 b^{2}}{24 a b}$

$∴ \frac{3 a}{4 b}, \frac{- 2 b}{3 a}$ are the roots of the given equation.

Find the roots of the equation: $12 a b x^{2} - (9 a^{2} - 8 b^{2}) x - 6 a b = 0$

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

Find the roots of the equation: 12abx2-(9a2-8b2)x-6ab=0

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

Find the roots of the equation: $12 a b x^{2} - (9 a^{2} - 8 b^{2}) x - 6 a b = 0$