Halp with Quadratics! (1 Viewer)

bachviete · Feb 13, 2010

ohexploitable said:
$Find\; x\; in\; terms\; of\; a\; and\; b\; if\; \frac{a^{2}b}{x^{2}}\; +\; \left ( 1\; +\; \frac{b}{x} \right )\; a\; =\; 2b\; +\; \frac{a^{2}}{x}.$

$Thanks!$

$a^2b + ax^2 + abx = 2bx^2 + a^2x$

$(ax^2 - 2bx^2) + (abx -a^2x) + a^2b = 0$

$(a-2b)x^2 + (ab - a^2)x +a^2b = 0$

$x = \frac{-(ab - a^2) \pm \sqrt{(ab - a^2)^2 - 4(a-2b)(a^2b)}}{2(a-2b)}$

$= \frac{a(a-b) \pm \sqrt{a^2(b-a)^2 - 4a^2(a-2b)(b)}}{2(a-2b)}$

$= \frac{a(a-b) \pm a \sqrt{(b-a)^2 - 4b(a-2b)}}{2(a-2b)}$

$= \frac{a(a-b) \pm a \sqrt{a^2 - 2ab + b^2 - 4ab + 8b^2}}{2(a-2b)}$

$= \frac{a(a-b) \pm a \sqrt{a^2 - 6ab + 9b^2}}{2(a-2b)}$

$= \frac{a(a-b) \pm a \sqrt{(a-3b)^2}}{2(a-2b)}$

$= \frac{a(a-b) \pm a(a-3b)}{2(a-2b)}$

$x = \frac{a(a-b) + a(a-3b)}{2(a-2b)}$

$= \frac{a(a-b + a-3b)}{2(a-2b)}$

$= \frac{a(2a - 4b)}{2a - 4b}$

$= a$

Also,

$x = \frac{a(a-b) - a(a-3b)}{2(a-2b)}$

$= \frac{a(a-b-a+3b)}{2(a-2b)}$

$= \frac{a(2b)}{2(a-2b)}$

$= \frac{ab}{a-2b}$

$\boxed{ \therefore x = a\, , \,\, \frac{ab}{a-2b}}$

ohexploitable · Feb 13, 2010

Thank you!

Halp with Quadratics! (1 Viewer)

ohexploitable

hey

bachviete

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ohexploitable

hey

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