Solve the polynomial equation
$9 x^{2} - 4 z^{2} - 24 x y + 16 y^{2} + 20 y - 15 x + 10$

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Solution

$9 x^{2} - 4 z^{2} - 24 x y + 16 y^{2} + 20 y - 15 x + 10 = 0$

$9 x^{2} - (24 y + 15) x - 4 z^{2} + 16 y^{2} + 20 y + 10 = 0$

$x_{1, 2} = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

$x = \frac{- (- 24 y - 15) \pm \sqrt{{(- 24 y - 15)}^{2} - 4 \cdot 9 (- 4 z^{2} + 16 y^{2} + 20 y + 10)}}{2 \cdot 9}$

$x = \frac{8 y \pm \sqrt{16 z^{2} - 15} + 5}{6}$

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Factorize: $9 x^{2} + 16 y^{2} + 4 z^{2} - 24 x y + 16 y z - 12 x z$

3 x - 4 y

9 x^{2} - 16 y^{2}

6 x^{2} - 8 y^{2}

9 x^{2} + 16 y^{2} + 24 x y

9 x^{2} + 16 y^{2} - 24 x y

The lines represented by the equation $9 x^{2} + 24 x y + 16 y^{2} + 21 x + 28 y + 6 = 0$ are

9 x^{2} - 24 x y + 16 y^{2} - 12 x + 16 y - 12 = 0

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