The sum of all the roots of the equation (x-1)^2-5|x-1|+6=0 , is:

Solution & Explanation

### Related Formula X^2 = |X|^2 Quadratic factorization: t^2 - 5t + 6 = (t-2)(t-3) ### Core Logic Rewrite the equation taking |x - 1| = t, where t geq 0. Since (x-1)^2 = |x-1|^2, the equation becomes: t^2 - 5t + 6 = 0 ### Step 1: Solve for modulus (t - 2)(t - 3) = 0 Rightarrow t = 2, 3 Since both roots are positive, both provide valid solutions for the modulus. |x - 1| = 2 quad textand quad |x - 1| = 3 ### Step 2: Unpack x values From |x - 1| = 2: x - 1 = 2 Rightarrow x = 3 x - 1 = -2 Rightarrow x = -1 From |x - 1| = 3: x - 1 = 3 Rightarrow x = 4 x - 1 = -3 Rightarrow x = -2 The roots are 3, -1, 4, -2. ### Step 3: Sum of Roots textSum = 3 + (-1) + 4 + (-2) = 4 ### Pattern Recognition In symmetric modulus equations f(|x-a|) = 0, every valid root t generates twin solutions (a+t) and (a-t). The sum of each pair is perfectly 2a. If there are n distinct valid positive roots for t, the sum of all x-roots is exactly n times 2a. Here, n=2, a=1 Rightarrow 2 times 2(1) = 4. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Quadratic Equations

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Q20 jee_main_2024_31_jan_morning Sign of Quadratic Expressions
Let S be the set of positive integral values of a for which fracax^2 + 2(a + 1)x + 9a + 4x^2 - 8x + 32 < 0, forall x in mathbbR. Then, the number of elements in S is:
  • A. 1
  • B. 0
  • C. infty
  • D. 3

Solution

### Core Logic For the denominator x^2 - 8x + 32, D = 64 - 128 < 0 and a = 1 > 0. Thus, x^2 - 8x + 32 > 0 forall x in mathbbR. ### Step 1: Constraint on Numerator Since the denominator is always positive, the numerator must be strictly negative for all x in mathbbR. ax^2 + 2(a + 1)x + 9a + 4 < 0 quad forall x in mathbbR This requires a < 0 and D < 0. ### Step 2: Conclusion Since a must be strictly less than 0, there are no *positive* integral values of a that satisfy the condition. Hence, S is an empty set. Number of elements is 0. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Quadratic Equations

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