Question

The solution of the equation $$\sqrt{x^2-16}-(x-4)=\sqrt{x^2-5x+4}$$ is?

A
{4,5,133}
B
{4,5}
C
{4}
D
{5,133}

Solution

The correct option is A $$\left\{4, 5, -\dfrac{13}{3}\right\}$$$$\quad \sqrt{(x-4)(x+4)-(x-4)}=\sqrt{(x-4)(x-1)}$$$$\Rightarrow \sqrt{(x-4)}\{\sqrt{x+4}-\sqrt{x-4}\}=\sqrt{(x-4)} \times \sqrt{(x-1)}$$$$\Rightarrow \sqrt{x-4}=0 \quad \Rightarrow \quad x=4$$or $$\sqrt{x+4}-\sqrt{x-4}=\sqrt{x-1}$$Squaring both sides,$$x+4+(x-4)-2 \sqrt{x^{2}-16}=x-1$$$$\Rightarrow x+1=2 \sqrt{x^{2}-16}$$Squaring both sides we get$$\Rightarrow 3 x^{2}-2 x-65=0$$$$\Rightarrow x=5, \dfrac{-13}{3}$$$$\therefore x=\left\{5,4,-\dfrac{13}{3}\right\}$$option $$A$$ is correct.Mathematics

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