Multiple choice

The number of real roots of the equation: (x2 + 4x)2 - (x + 2)2 - 26 = 0 is

  1. 0

  2. 1

  3. 2

  4. 4
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Let y = x^2 + 4x. The equation is y^2 - (x+2)^2 - 26 = 0. This does not simplify easily. Instead, expand: (x^2+4x)^2 - (x^2+4x+4) - 26 = 0. Let u = x^2+4x. Then u^2 - u - 30 = 0. Factoring gives (u-6)(u+5) = 0. Case 1: x^2+4x-6=0 has discriminant 16 - 4(1)(-6) = 40 > 0 (2 roots). Case 2: x^2+4x+5=0 has discriminant 16 - 4(1)(5) = -4 < 0 (0 real roots). Total real roots = 2.

AI explanation

By substituting t = (x^2 + 4x) into the original equation (x^2 + 4x)^2 - (x + 2)^2 - 26 = 0, we rewrite the expression as t^2 - x^2 - 4x - 4 - 26 = 0. Replacing x^2 + 4x with t yields the quadratic equation t^2 - t - 30 = 0. Factoring this gives (t - 6)(t + 5) = 0, meaning t = 6 or t = -5, so x^2 + 4x = 6 has two real roots while x^2 + 4x = -5 has a discriminant of 16 - 20 = -4 and no real roots. The equation therefore has exactly 2 real roots.

Keep practicing — related questions