Question

The number of solutions of the equation \(9x^2 - 18|x| + 5 = 0\) belonging to the domain of definition of \(\log_e \{(x+1)(x+2)\}\), is

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Always check domain restrictions after solving the equation.

Answer 1

The Correct Option is C

To solve the problem, we need to determine the number of solutions for the equation \(9x^2 - 18|x| + 5 = 0\) that satisfy the domain of definition for \(\log_e \{(x+1)(x+2)\}\).

The domain of \(\log_e \{(x+1)(x+2)\}\) is determined by the condition that the argument must be positive:

1. \({(x+1)(x+2) > 0}\)

This inequality can be solved by analyzing the sign changes around its roots.

• Roots are \(x = -1\) and \(x = -2\).
• Check intervals: \((-\infty, -2)\), \((-2, -1)\), \((-1, \infty)\).
• In \((-\infty, -2)\), both factors \((x+1)\) and \((x+2)\) are negative, so their product is positive.
• In \((-2, -1)\), \((x+2)\) is positive, but \((x+1)\) is negative, so their product is negative.
• In \((-1, \infty)\), both factors \((x+1)\) and \((x+2)\) are positive, so their product is positive.

Next, solve the equation \(9x^2 - 18|x| + 5 = 0\).

The equation is quadratic in terms of \(|x|\). Let \(y = |x|\), then:

1. \(9y^2 - 18y + 5 = 0\)

Using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 9\), \(b = -18\), \(c = 5\):

1. \(y = \frac{18 \pm \sqrt{(-18)^2 - 4 \cdot 9 \cdot 5}}{18}\) \(y = \frac{18 \pm \sqrt{324 - 180}}{18}\) \(y = \frac{18 \pm \sqrt{144}}{18}\) \(y = \frac{18 \pm 12}{18}\)

Thus, \(y = 1.666\) or \(y = 0.333\).

Convert back: \(|x| = 1.666\) or \(|x| = 0.333\).

Find the corresponding \(x\) values:

• For \(|x| = 1.666\), \(x = \pm 1.666\).
• For \(|x| = 0.333\), \(x = \pm 0.333\).
• \(x = -1.666\) lies within \((-2, -1)\) which does not satisfy the domain.
• \(x = 1.666\), \(-0.333\), and \(0.333\) all satisfy the domain.

Thus, the number of solutions from the domain is \(x = 1.666, -0.333, 0.333\).

Therefore, the correct answer is:

3