Problem 64 - Algebra and Trigonometry

Question

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ (x+1)^{4}+16=0 $$

Step-by-Step Solution

Answer

No real solutions exist for the equation.

Step 1: Identify the Equation

The given equation is \((x+1)^{4} + 16 = 0\). Our task is to solve this equation and find all real solutions.

Step 2: Isolate the Polynomial

Move the constant term to the other side of the equation: \((x+1)^{4} = -16\).

Step 3: Analyze the Fourth Power

Note that raising any real number to the power of 4 will always result in a non-negative number. Therefore, a real number raised to the power of 4 cannot equal \(-16\).

Step 4: Conclude the Solution

Since there is no real number that can satisfy \((x+1)^{4} = -16\), the equation has no real solutions.

Key Concepts

Real Solutions

When we talk about finding the real solutions of an equation, we're essentially looking for the values that make the equation true using real numbers. Real numbers include all the numbers on the number line, such as

• integers (e.g., -3, 0, 4)
• fractions (e.g., 1/2, -7/4)
• irrational numbers (e.g., \(\pi, \sqrt{2}\))

In our exercise, we're given the equation \[ (x+1)^{4} + 16 = 0 \]This means our job is to determine if there's any real number that can satisfy this condition. Most often, this involves manipulating the equation, isolating terms, and checking if the expression results in a valid real number. In this specific case, the process led us to a situation where it wasn't possible to find such a number, leading to the conclusion of no real solutions.

Fourth Power

The fourth power of a number means raising it to the power of four, which is essentially multiplying the number by itself three additional times. For any real number \( a \), the fourth power, or \( a^4 \), is always non-negative.This is because:

• If \( a \) is positive, \( a^4 \) is positive since a positive number raised to any power remains positive.
• If \( a \) is zero, \( a^4 \) is zero.
• If \( a \) is negative, \( a^4 \) becomes positive because multiplying four negative numbers results in a positive product.

Therefore, when we try to solve \[(x+1)^4 = -16\],we find that the expression \((x+1)^4\) can never be negative. Hence, it becomes clear why there are no real solutions to this equation.

Negative Number

Negative numbers are numbers less than zero, signified by a minus (-) sign. In the context of our equation, they play a critical role. The equation \[ (x+1)^{4} = -16 \]presents a contradiction since, as discussed in the fourth power section, a positive number cannot equate to a negative number in the real number domain. When considering powers, specifically even powers like four, the result is always non-negative due to repeated multiplication. Thus, the impossibility of equality between a non-negative fourth power and a negative number like -16 becomes apparent, providing clarity on why real solutions do not exist in this particular scenario.