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CAT 2023 Slot 1 — QA Question 6

Solving Quadratic EquationsEasy

The number of integral solutions of equation 2|x|(x2 + 1) = 5x2 is?

Answer & solution

Answer: 3

Solution

Easy

The ∣x∣|x| makes this two cases (x≥0x\ge 0 and x<0x<0). In each case the equation becomes a cubic that factors, and only the integer roots count. Collect the distinct integers across both cases.

1

Case x≥0x\ge 0 (∣x∣=x|x|=x):

2x(x2+1)=5x2 ⇒ x(2x2−5x+2)=0 ⇒ x(2x−1)(x−2)=02x(x^2+1)=5x^2\ \Rightarrow\ x\big(2x^2-5x+2\big)=0\ \Rightarrow\ x(2x-1)(x-2)=0 ⇒ x=0, 12, 2\Rightarrow\ x=0,\ \tfrac12,\ 2

Integers here: x=0x=0 and x=2x=2 (discard 12\tfrac12).

2

Case x<0x<0 (∣x∣=−x|x|=-x):

−2x(x2+1)=5x2 ⇒ x(2x2+5x+2)=0 ⇒ x(2x+1)(x+2)=0-2x(x^2+1)=5x^2\ \Rightarrow\ x\big(2x^2+5x+2\big)=0\ \Rightarrow\ x(2x+1)(x+2)=0 ⇒ x=0, −12, −2\Rightarrow\ x=0,\ -\tfrac12,\ -2

Integers here: x=−2x=-2 (and x=0x=0, already counted; discard −12-\tfrac12).

3

Collect distinct integer solutions:

{−2, 0, 2} ⇒ 3 values\{-2,\ 0,\ 2\}\ \Rightarrow\ 3\ \text{values}
3 integral solutions\mathbf{3}\ \text{integral solutions}

The equation 2∣x∣(x2+1)=5x22|x|(x^2+1)=5x^2 is even in xx (replacing xx by −x-x leaves it unchanged), so non-zero roots come in ±\pm pairs. Find x=2x=2 works for x>0x>0, add its mirror x=−2x=-2, throw in x=0x=0, and you have 33.

CAT 2023 Slot 1 QA Q6: The number of integral solutions of equation 2|x|(x 2 + 1) = 5x 2 is? — Solution | TheCATExam