Question

The number of solutions to the equation | x | (6x² + 1) = 5x² is

• A. 7
• B. 9
• C. 8
• D. 5

Answer 1

The Correct Option is D

Scenario I: \( x = 0 \)

Clearly, \[ |0|(6 \cdot 0^2 + 1) = 5 \cdot 0^2 \Rightarrow 0 = 0 \] So, \( x = 0 \) is a valid solution.

Scenario II: \( x > 0 \Rightarrow |x| = x \)

Substitute into the equation: \[ x(6x^2 + 1) = 5x^2 \Rightarrow 6x^3 + x = 5x^2 \Rightarrow 6x^3 - 5x^2 + x = 0 \Rightarrow x(6x^2 - 5x + 1) = 0 \]

Solving the quadratic: \[ 6x^2 - 5x + 1 = 0 \Rightarrow x = \frac{1}{2}, \frac{1}{3} \] Both are positive, so both are valid solutions.

Scenario III: \( x < 0 \Rightarrow |x| = -x \)

Substitute into the equation: \[ (-x)(6x^2 + 1) = 5x^2 \Rightarrow -6x^3 - x = 5x^2 \Rightarrow 6x^3 + 5x^2 + x = 0 \Rightarrow x(6x^2 + 5x + 1) = 0 \]

Solving the quadratic: \[ 6x^2 + 5x + 1 = 0 \Rightarrow x = -\frac{1}{2}, -\frac{1}{3} \] Both are negative and valid.

Final Answer:

We have 5 valid solutions: \[ \boxed{0, \frac{1}{2}, \frac{1}{3}, -\frac{1}{2}, -\frac{1}{3}} \]