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Quadratic Equation and Inequalities question

2019 · 10 Apr · Shift 2 · Q25
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  5. /2019 · 10 Apr · Shift 2 · Q25

Quadratic Equation and Inequalities question

2019 · 10 Apr · Shift 2 · Q25

JEE MainMathematicsQuadratic Equation and InequalitiesMCQ+4 / −1
The number of real roots of the equation 5 + |2x – 1| = 2x (2x – 2) is
  1. A
    2
  2. B
    1
  3. C
    3
  4. D
    4
View written solutionFree

Correct answer: A

  1. Given equation

    5+∣2x−1∣=2x(2x−2)5 + |2x-1| = 2x(2x-2)5+∣2x−1∣=2x(2x−2)

    Simplify the right-hand side:

    2x(2x−2)=4x2−4x2x(2x-2)=4x^2-4x2x(2x−2)=4x2−4x

    So the equation becomes

    5+∣2x−1∣=4x2−4x5+|2x-1|=4x^2-4x5+∣2x−1∣=4x2−4x

  2. Split into cases because of the modulus

    The expression ∣2x−1∣|2x-1|∣2x−1∣ changes at

    2x−1=0  ⟹  x=122x-1=0 \implies x=\frac122x−1=0⟹x=21​

    So we consider:

    • Case 1: x≥12x\ge \frac12x≥21​
    • Case 2: x<12x<\frac12x<21​

  1. Case 1: x≥12x\ge \frac12x≥21​

    Then

    ∣2x−1∣=2x−1|2x-1|=2x-1∣2x−1∣=2x−1

    Substitute:

    5+(2x−1)=4x2−4x5+(2x-1)=4x^2-4x5+(2x−1)=4x2−4x

    4+2x=4x2−4x4+2x=4x^2-4x4+2x=4x2−4x

    4x2−6x−4=04x^2-6x-4=04x2−6x−4=0

    Divide by 222:

    2x2−3x−2=02x^2-3x-2=02x2−3x−2=0

    Factor:

    (2x+1)(x−2)=0(2x+1)(x-2)=0(2x+1)(x−2)=0

    Hence,

    x=−12,  2x=-\frac12,\; 2x=−21​,2

    But in this case we need x≥12x\ge \frac12x≥21​, so only

    x=2x=2x=2

    is valid.

  1. Case 2: x<12x<\frac12x<21​

    Then

    ∣2x−1∣=−(2x−1)=1−2x|2x-1|=-(2x-1)=1-2x∣2x−1∣=−(2x−1)=1−2x

    Substitute:

    5+(1−2x)=4x2−4x5+(1-2x)=4x^2-4x5+(1−2x)=4x2−4x

    6−2x=4x2−4x6-2x=4x^2-4x6−2x=4x2−4x

    4x2−2x−6=04x^2-2x-6=04x2−2x−6=0

    Divide by 222:

    2x2−x−3=02x^2-x-3=02x2−x−3=0

    Factor:

    (2x−3)(x+1)=0(2x-3)(x+1)=0(2x−3)(x+1)=0

    Hence,

    x=32,  −1x=\frac32,\; -1x=23​,−1

    But in this case we need x<12x<\frac12x<21​, so only

    x=−1x=-1x=−1

    is valid.

  1. Total real roots

    The valid real roots are:

    x=2,  −1x=2,\; -1x=2,−1

    Therefore, the number of real roots is

    222

  2. Option check

    • A: 222 ✅
    • B: 111 ❌
    • C: 333 ❌
    • D: 444 ❌

Therefore, the correct option is A.

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