Solve Algebraically (x-3)^4-5=11

Solve Algebraically (x-3)^4-5=11

When approaching algebraic equations such as (x-3)^4 - 5 = 11, the goal is to isolate the variable—in this case, x—and find its value(s). This process involves understanding the properties of exponents, applying inverse operations, and carefully handling the algebraic manipulations to arrive at the solution. Solving equations like this can sometimes produce more than one solution, especially when higher powers are involved, so it’s essential to consider all possible roots, including negative ones, when applicable. In this article, we will explore step-by-step methods to solve the equation (x-3)^4 - 5 = 11, discuss potential challenges, and analyze what the solutions tell us about the problem.

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Understanding the Equation

Equation Breakdown

The given equation is:

( x - 3 )^4 - 5 = 11

This expression involves:

• A binomial expression raised to the fourth power: (x - 3)^4


• A constant subtraction: -5


• An equal sign with a constant on the right side: 11

The first step in solving such an equation is to understand what each part represents and how they relate to each other. The key is to isolate the term with the variable first, then work backwards through the operations.

Goal of the Solution

• Isolate the exponential expression (x - 3)^4.
• Solve for the inner expression (x - 3).
• Find the value(s) of x.

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Step-by-Step Solution Process

Step 1: Isolate the Power Term

To start, add 5 to both sides of the equation to isolate the exponential term:

( x - 3 )^4 - 5 + 5 = 11 + 5

Simplifies to:

( x - 3 )^4 = 16

Step 2: Take the Fourth Root of Both Sides

Next, to solve for (x - 3), take the fourth root of both sides. Recall that:

• The fourth root is the same as raising to the power of 1/4.
• When taking roots of both sides, consider all possible roots, including positive and negative, because (x - 3)^4 is always non-negative, but the fourth root can be both positive and negative.

Applying the fourth root:

x - 3 = ±√[4]{16}

Since 16 is a perfect fourth power (2^4 = 16), the fourth roots are:

x - 3 = ±2

Note: The ± indicates two solutions: one positive and one negative.

Step 3: Solve for x

Now, solve for x in both cases:

1. x - 3 = 2

x = 2 + 3

x = 5

1. x - 3 = -2

x = -2 + 3

x = 1

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Final Solutions and Verification

Solutions Summary

The solutions to the original equation are:

• x = 5
• x = 1

Verification of Solutions

It’s important to verify these solutions by substituting them back into the original equation:

For x = 5:

• Compute (5 - 3)^4 - 5


• (2)^4 - 5 = 16 - 5 = 11


• Which matches the right side of the original equation.

For x = 1:

• Compute (1 - 3)^4 - 5


• (-2)^4 - 5 = 16 - 5 = 11


• Which also satisfies the original equation.

Since both solutions satisfy the original equation, they are valid.

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Discussion of the Solution and Additional Considerations

Impact of Even Powers on Solutions

Because the power involved is even (the fourth power), both positive and negative roots of the intermediate step are valid solutions. This is a common characteristic in equations involving even exponents, where negative inputs can yield positive outputs.

Potential for Extraneous Solutions

In some algebraic equations, operations such as taking even roots can introduce extraneous solutions, which are solutions that satisfy the intermediate steps but not the original equation. However, in this case, verification confirms that both solutions (x = 1 and x = 5) are valid.

Extensions and Related Problems

• What happens if the right side is a different constant? The process remains similar, but the roots and solutions may differ.
• How to handle equations involving higher even powers or different exponents.
• The importance of verifying solutions, especially when dealing with roots and powers.

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Summary and Key Takeaways

• Start by isolating the exponential or polynomial expression involving the variable.
• When dealing with even powers, consider both positive and negative roots.
• Take the appropriate root (in this case, the fourth root) to simplify the equation.
• Solve for the variable by isolating it after applying roots.
• Always verify solutions to ensure they satisfy the original equation.
• Recognize that algebraic equations involving even powers often have multiple solutions due to symmetry.

Conclusion

Solving algebraically the equation (x - 3)^4 - 5 = 11 involves a straightforward sequence of algebraic steps: isolating the power term, taking roots, and solving for x. The process highlights the importance of understanding how exponents and roots interact and the necessity of verifying solutions. The solutions x = 1 and x = 5 demonstrate how symmetry in even powers can produce multiple valid solutions. Mastering these techniques enhances problem-solving skills and deepens understanding of algebraic concepts, providing a foundation for tackling more complex equations in mathematics.