An algebra problem by xi guan

Algebra Level 3

Find all the possible values of x x satisfying x 2 − 17 x − 2 − x 2 − 17 x − 22 = 2 \sqrt{x^2-17x-2} - \sqrt{x^2-17x-22} = 2 .

19 and -2 -19 and 2 19 19 and 2 2

Xi Guan by xi guan , 20, Taiwan

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1 solution

x 2 − 17 x − 2 − x 2 − 17 x − 22 = 2 x 2 − 17 x − 12 + 10 − x 2 − 17 x − 12 − 10 = 2 Let u = x 2 − 17 x − 12 u + 10 − u − 10 = 2 Squaring both sides u + 10 − 2 u 2 − 1 0 2 + u − 10 = 4 Divide by 2 both sides and rearrange. u − 2 = u 2 − 100 Squaring both sides u 2 − 4 u + 4 = u 2 − 100 4 u = 104 ⟹ u = 26 Put back u = x 2 − 17 x − 12 x 2 − 17 x − 12 = 26 x 2 − 17 x − 38 = 0 ( x − 19 ) ( x + 2 ) = 0 ⟹ x = 19 and − 2 \begin{aligned} \sqrt{x^2-17x-2} - \sqrt{x^2-17x-22} & = 2 \\ \sqrt{{\color{#3D99F6}x^2-17x-12}+10} - \sqrt{{\color{#3D99F6}x^2-17x-12}-10} & = 2 & \small \color{#3D99F6} \text{Let }u = x^2-17x-12 \\ \sqrt{{\color{#3D99F6}u}+10} - \sqrt{{\color{#3D99F6}u}-10} & = 2 & \small \color{#3D99F6} \text{Squaring both sides} \\ u+10 - 2\sqrt{u^2-10^2} + u -10 & = 4 & \small \color{#3D99F6} \text{Divide by 2 both sides and rearrange.} \\ u - 2 & = \sqrt{u^2-100} & \small \color{#3D99F6} \text{Squaring both sides} \\ u^2 - 4u + 4 & = u^2 - 100 \\ 4u & = 104 \\ \implies \color{#3D99F6} u & = 26 & \small \color{#3D99F6} \text{Put back }u = x^2-17x-12 \\ \color{#3D99F6} x^2-17x - 12 & = 26 \\ x^2-17x-38 & = 0 \\ (x-19)(x+2) & = 0 \\ \implies x & = \boxed{19 \text{ and }-2} \end{aligned}

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Maybe the fastest way is substituting all values given.

. . - 2 months, 3 weeks ago

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