Show that the statement .
p : 'If x is a real number such that `x^(3)+ 4x=0,` then x=0' is true by
Method of contrapositive

Answer

To prove the statement \( p: \) "If \( x \) is a real number such that \( x^3 + 4x = 0 \), then \( x = 0 \)" is true using the method of contrapositive, we can follow these steps:

### Step 1: Understand the statement and its contrapositive
The original statement \( p \) can be expressed in the form of implications:
- Let \( q: x^3 + 4x = 0 \)
- Let \( r: x = 0 \)

The statement \( p \) can be rewritten as \( q \implies r \).

The contrapositive of this statement is:
- If \( x \neq 0 \) (negation of \( r \)), then \( x^3 + 4x \neq 0 \) (negation of \( q \)).

### Step 2: Assume the negation of \( r \)
Assume \( r \) is false, which means:
- \( x \neq 0 \)

### Step 3: Analyze the equation \( x^3 + 4x \)
We need to show that under the assumption \( x \neq 0 \), the equation \( x^3 + 4x \) does not equal zero.

Factor the expression:
\[ x^3 + 4x = x(x^2 + 4) \]

### Step 4: Determine the conditions for \( x^3 + 4x \)
1. Since \( x \neq 0 \), the term \( x \) is non-zero.
2. The term \( x^2 + 4 \) is always positive because:
- \( x^2 \) is non-negative for all real \( x \).
- Adding 4 ensures that \( x^2 + 4 > 0 \) for all real \( x \).

### Step 5: Conclude that \( x^3 + 4x \neq 0 \)
Since both factors \( x \) and \( x^2 + 4 \) are non-zero when \( x \neq 0 \):
\[ x(x^2 + 4) \neq 0 \]
Thus, we conclude that:
\[ x^3 + 4x \neq 0 \]

### Step 6: State the conclusion
Since we have shown that if \( x \neq 0 \), then \( x^3 + 4x \neq 0 \), we have proved the contrapositive:
\[ \text{If } x \neq 0, \text{ then } x^3 + 4x \neq 0 \]
This confirms that the original statement \( p \) is true.

### Summary
We have shown that the statement \( p \) is true by proving its contrapositive.

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