If-a-0-then-the-real-values-of-x-satisfying-x-2-2a-x-a-3a-2-0-are-

Question Number 5638 by Daily last updated on 23/May/16

If a ⩽ 0, then the real values of x

satisfying x^{2} - 2 a ∣ x - a ∣ - 3 a^{2} = 0 are

Answered by Yozzii last updated on 23/May/16

W e c a n l o o k f o r s o l u t i o n s b a s e d o n

i n t e r v a l s t h a t c o v e r t h e e n t i r e r e a l l i n e .

a ⩽ 0 i s s o m e f i x e d r e a l v a l u e .

I f x - a > 0 o r x > a \Rightarrow∣ x - a ∣= x - a .

∴ x^{2} - 2 a (x - a) - 3 a^{2} = 0

x^{2} - 2 a x - a^{2} = 0

x^{2} - 2 a x + a^{2} - 2 a^{2} = 0

{(x - a)}^{2} = 2 a^{2}

\Rightarrow x = a \pm \sqrt{2} \sqrt{a^{2}}

x = a \pm \sqrt{2} ∣ a ∣ w h i c h a r e r e a l v a l u e s o f x .

a ⩽ 0 \Rightarrow∣ a ∣= - a \Rightarrow x = (1 \pm \sqrt{2}) a

N o w x > a \Rightarrow x \neq (1 + \sqrt{2}) a b u t x = (1 - \sqrt{2}) a

s i n c e a ⩽ 0 . {a ⩽ 0 \Rightarrow (1 - \sqrt{2}) a ⩾ 0 ⩾ a o r x ⩾ a}

I f x = a ⩽ 0 t h e e q u a t i o n g i v e s

a^{2} - 3 a^{2} = 0 \Rightarrow 2 a^{2} = 0 . T h i s i s t r u e o n l y i f a = 0 .

∴ x = 0 i s p o s s i b l e .

I f x - a < 0 \Rightarrow x < a \Rightarrow∣ x - a ∣= a - x .

∴ x^{2} - 2 a (a - x) - 3 a^{2} = 0

x^{2} + 2 a x - 5 a^{2} = 0

x^{2} + 2 a x + a^{2} - 6 a^{2} = 0

{(x + a)}^{2} = 6 a^{2}

\Rightarrow x = - a \pm \sqrt{6} ∣ a ∣= a (- 1 \pm \sqrt{6})

S i n c e a ⩽ 0 a n d x < a \Rightarrow x \neq (- 1 - \sqrt{6}) a

b u t x = a (\sqrt{6} - 1) .

∴ I f a = 0 t h e q u a d r a t i c e q u a t i o n h a s

t w o e q u a l r o o t s t h e n x = 0 . I f a \neq 0 t h e n

x = (\sqrt{6} - 1) a o r x = (1 - \sqrt{2}) a .

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