Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
$12 a b x^{2} - (9 a^{2} - 8 b^{2}) x - 6 a b = 0, w h e r e a \neq 0 a n d b \neq 0$

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Solution

Using quadratic formula
$12 a b x^{2} - (9 a^{2} - 8 b^{2}) x - 6 a b = 0 A = 12 a b B = - (9 a^{2} - 8 b^{2}) C = - 6 a b$

∴Discriminant, $D = B^{2} - 4 A C = (- (9 a^{2} - 8 b^{2}))^{2} - 4 \times (12 a b) \times (- 6 a b) = 81 a^{4} + 144 a^{2} b^{2} + 64 b^{4} = (9 a^{2} + 8 b^{2})^{2} \geq 0$

As $D \geq 0$ therefore, the roots are real.

$x = \frac{- B \pm \sqrt{D}}{2 A} = \frac{- (- (9 a^{2} - 8 b^{2})) \pm \sqrt{(9 a^{2} + 8 b^{2})^{2}}}{2 (12 a b)} = \frac{(9 a^{2} - 8 b^{2}) \pm (9 a^{2} + 8 b^{2})}{24 a b} = \frac{(9 a^{2} - 8 b^{2}) + (9 a^{2} + 8 b^{2})}{24 a b} o r \frac{(9 a^{2} - 8 b^{2}) - (9 a^{2} + 8 b^{2})}{24 a b} = \frac{9 a^{2} - 8 b^{2} + 9 a^{2} + 8 b^{2}}{24 a b} o r \frac{9 a^{2} - 8 b^{2} - 9 a^{2} - 8 b^{2}}{24 a b} = \frac{18 a^{2}}{24 a b} o r \frac{- 16 b^{2}}{24 a b} = \frac{3 a}{4 b} o r \frac{- 2 b}{3 a}$

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Similar questions

Solve each of the following quadratic equations:
$12 a b x^{2} - (9 a^{2} - 8 b^{2}) x - 6 a b = 0$

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

, where a

\neq

0 and b

\neq

Q. Evaluate

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

, where

a \neq 0

and

b \neq 0

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

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
$3 a^{2} x^{2} + 8 a b x + 4 b^{2} = 0, a \neq 0$

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Find the roots of each of the following equations, if they exist, by applying the quadratic formula: 12abx2−(9a2−8b2)x−6ab=0, where a≠0 and b≠0

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
$12 a b x^{2} - (9 a^{2} - 8 b^{2}) x - 6 a b = 0, w h e r e a \neq 0 a n d b \neq 0$