Question

The number of real solutions of the equation\[x \left( x^2 + 3|x| + 5|x - 1| + 6|x - 2| \right) = 0\]is ______.

Answer 1

Correct Answer: 1

Analyze the equation:

\[ x(x^2 + 3x) + |x - 1| + 6|k - 2| = 0 \implies x^3 + 3x^2 + |x - 1| + C = 0. \]

Substitute \( C = 6|k - 2| \):

\[ x^3 + 3x^2 + |x - 1| + C = 0. \]

Consider cases for absolute value:

Case 1: \( x \geq 1 \):

\[ x^3 + 3x^2 + (x - 1) + C = 0 \implies x^3 + 3x^2 + x + (C - 1) = 0. \]

Case 2: \( x < 1 \):

\[ x^3 + 3x^2 + (-x + 1) + C = 0 \implies x^3 + 3x^2 - x + (C + 1) = 0. \]

Finding the number of real solutions: The cubic functions yield at most one real root due to their monotonically increasing nature.

Evaluate the function: Since the cubic functions are monotonic, we conclude: There is one real solution in each case.
Thus, the number of real solutions is: 1.

Answer 2

The problem asks for the number of real solutions to the equation \(x \left( x^2 + 3|x| + 5|x - 1| + 6|x - 2| \right) = 0\).

Concept Used:

1. Zero-Product Property: The equation \( A \cdot B = 0 \) is true if and only if \( A = 0 \) or \( B = 0 \) (or both).

2. Properties of Absolute Value and Squares: For any real number \( a \), the following properties hold:

• \( |a| \geq 0 \)
• \( a^2 \geq 0 \)

A sum of non-negative terms is equal to zero if and only if each term in the sum is individually equal to zero.

Step-by-Step Solution:

Step 1: Apply the Zero-Product Property to the given equation.

The equation is in the form \( A \cdot B = 0 \), where:

\[ A = x \] \[ B = x^2 + 3|x| + 5|x - 1| + 6|x - 2| \]

According to the zero-product property, the solutions can be found by setting each factor to zero. So, either \( A = 0 \) or \( B = 0 \).

Step 2: Analyze the first case, \( A = 0 \).

\[ x = 0 \]

This gives one immediate real solution.

Step 3: Analyze the second case, \( B = 0 \).

\[ x^2 + 3|x| + 5|x - 1| + 6|x - 2| = 0 \]

Step 4: Examine the individual terms of the expression in \( B \).

For any real number \( x \):

• The term \( x^2 \) is always non-negative, i.e., \( x^2 \geq 0 \).
• The term \( 3|x| \) is always non-negative, i.e., \( 3|x| \geq 0 \).
• The term \( 5|x - 1| \) is always non-negative, i.e., \( 5|x - 1| \geq 0 \).
• The term \( 6|x - 2| \) is always non-negative, i.e., \( 6|x - 2| \geq 0 \).

Step 5: Determine if the sum of these terms can be zero.

The expression is a sum of four non-negative terms. For the sum to be zero, each individual term must be zero simultaneously.

\[ x^2 = 0 \implies x = 0 \] \[ 3|x| = 0 \implies x = 0 \] \[ 5|x - 1| = 0 \implies x - 1 = 0 \implies x = 1 \] \[ 6|x - 2| = 0 \implies x - 2 = 0 \implies x = 2 \]

For the expression to be zero, we need \( x = 0 \), \( x = 1 \), and \( x = 2 \) to be true at the same time, which is impossible. Therefore, there is no real value of \( x \) for which the expression \( B \) is equal to zero. In fact, for any real \( x \), the expression \( B \) is always strictly positive. For example, if we test the potential solution \( x=0 \), the expression becomes \( 0^2 + 3|0| + 5|0-1| + 6|0-2| = 0 + 0 + 5 + 12 = 17 \neq 0 \).

Final Computation & Result:

The first factor, \( x \), gives the solution \( x = 0 \).

The second factor, \( x^2 + 3|x| + 5|x - 1| + 6|x - 2| \), can never be zero for any real \( x \).

Therefore, the only real solution to the entire equation is \( x = 0 \).

The number of real solutions is 1.