Source: Extra Practice

Find the number of real solutions for the equation x25x+6=0x^2 - 5|x| + 6 = 0.

Options

Option A is correct

4

Option B

2

Option C

0

Option D

3

Explanation

Since x² is same as |x|², we can rewrite the equation as |x|² - 5|x| + 6 = 0. Let y = |x|. Then y² - 5y + 6 = 0. Factoring gives (y - 2)(y - 3) = 0, so y = 2 or y = 3. Since y = |x|, we have |x| = 2 (which gives x = ±2) and |x| = 3 (which gives x = ±3). Total 4 real solutions: {2, -2, 3, -3}. Option B is wrong as it ignores negative roots. Option C and D are incorrect.