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CA Foundation · Quantitative Aptitude · Equations

The equation x⁴ − 5x² + 4 = 0 is solved by substituting y = x². How many distinct real values of x satisfy it, and what is the sum of their squares?

There are 4 real values, namely ±1 and ±2, and the sum of their squares is 10. Substituting y = x² gives y² − 5y + 4 = 0, so y = 1 or 4. Each y gives two x values, so squares add to 1+1+4+4 = 10.

  1. A4 real values; sum of squares 10Correct
  2. B2 real values; sum of squares 5
  3. C4 real values; sum of squares 5
  4. D4 real values; sum of squares 0

Explanation

Let y = x²: y² − 5y + 4 = 0, giving y = 1 or y = 4. So x = ±1 or ±2, which are 4 distinct real values. Sum of squares = 1 + 1 + 4 + 4 = 10. The option giving 5 counts each y only once, missing the negative roots.

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