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JEE MainMathematicsQuadratic Equation

If the equation (2+ 3 )^ x^2-2x+a + (2- 3 )^ x^2-2x+a = 14 has exactly 3 distinct real solutions for x , then the value of a+5 is equal to

Options

  1. A8
  2. B5
  3. C6
  4. D4

Correct answer

D. 4

Step-by-step solution

Let u = (2+ 3 )^ x^2-2x+a . Since (2+ 3 )(2- 3 ) = 4 - 3 = 1 , we have (2- 3 ) = 1 2+ 3 . The given equation can be written as: u + 1 u = 14 u^2 - 14u + 1 = 0 u = 14 196 - 4 2 = 7 4 3 We can observe that (2 3 )^2 = 4 + 3 4 3 = 7 4 3 . Therefore, u = (2 3 )^2 . This implies that (2+ 3 )^ x^2-2x+a = (2+ 3 )^2 or (2+ 3 )^ x^2-2x+a = (2- 3 )^2 = (2+ 3 )⁻² . Equating the exponents, we get two quadratic equations: x^2 - 2x + a = 2 x^2 - 2x + (a - 2) = 0 (Equation 1) x^2 - 2x + a = -2 x^2 - 2x + (a + 2) = 0 (Equation 2) F

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