The sum of the solutions of the equation $|\sqrt{x}-2|+\sqrt{x}(\sqrt{x}-4)+2=0,(x>0)$ is equal to:

JEE Main · 2019 · 08 Apr 2019 · Shift 1 · Basic Mathematics

The sum of the solutions of the equation

∣ x - 2∣ + x (x - 4) + 2 = 0, (x > 0)

is equal to:

Answer: (C)

Notice the quadratic hidden in

x

: since

(x)^{2} - 4 x + 4 = (x - 2)^{2}

, the equation

∣ x - 2∣ + (x)^{2} - 4 x + 2 = 0

becomes

∣ x - 2∣ + (x - 2)^{2} - 2 = 0

. Put

t = ∣ x - 2∣

, so

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

and

t^{2} + t - 2 = 0

, factoring as

(t + 2) (t - 1) = 0

; only

t = 1

is admissible. Then

∣ x - 2∣ = 1

gives

x = 3

x = 1

, so

x = 9

x = 1

, and the solutions sum to

10

JEE Main · 2019 · 08 Apr 2019 · Shift 1 · Single correct option