Show that x-bc-ad is a solution of the quadratic equation ad2 ax-b-2c-d x-bc2-0-

Ad^2 ( ax/b+2c/d)x+bc^2=0 ⇒a^2d^2x^2/b+2cad^2x/d+bc^2=0 ⇒a^2d^2x^2/b+2cadx+bc^2=0 ⇒ a^2d^2x^2+2abcdx+b^2c^2=0 ⇒ (adx+bc)^2=0 ⇒ (adx+bc)=0 ⇒ x = bc/adx=0 ...

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Standard X

Mathematics

Roots of a Quadratic Equation

Show that x=-...

Question

Show that $x = - \frac{b c}{a d}$ is a solution of the quadratic equation $a d^{2} (\frac{a x}{b} + \frac{2 c}{d}) x + b c^{2} = 0$ .

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Solution

$a d^{2} (\frac{a x}{b} + \frac{2 c}{d}) x + b c^{2} = 0$

$\Rightarrow \frac{a^{2} d^{2} x^{2}}{b} + \frac{2 c a d^{2} x}{d} + b c^{2} = 0$

$\Rightarrow \frac{a^{2} d^{2} x^{2}}{b} + 2 c a d x + b c^{2} = 0$

$\Rightarrow a^{2} d^{2} x^{2} + 2 a b c d x + b^{2} c^{2} = 0$

$\Rightarrow (a d x + b c)^{2} = 0$

$\Rightarrow (a d x + b c) = 0$

$\Rightarrow x = \frac{- b c}{a d x} = 0$

hence proved

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Show that x=−bcad is a solution of the quadratic equation ad2(axb+2cd)x+bc2=0.

ad2(axb+2cd)x+bc2=0 ⇒a2d2x2b+2cad2xd+bc2=0 ⇒a2d2x2b+2cadx+bc2=0 ⇒a2d2x2+2abcdx+b2c2=0 ⇒(adx+bc)2=0 ⇒(adx+bc)=0 ⇒x=−bcadx=0 hence proved

Show that $x = - \frac{b c}{a d}$ is a solution of the quadratic equation $a d^{2} (\frac{a x}{b} + \frac{2 c}{d}) x + b c^{2} = 0$ .