Not dividing both sides of the equation by 16e

Oct 13, 2025 by windailys.com 7 views

Solving for f in the Equation d = 16ef^2

Understanding the Given Equation

The given equation is d = 16ef^2, where d, e, and f are variables. To solve for f, we need to isolate f on one side of the equation. This involves using algebraic manipulations to simplify the equation and express f in terms of the other variables.

Option A: f = ± 4√(d/e)

Let's start by examining option A: f = ± 4√(d/e). To verify this option, we can substitute the expression for f into the original equation and see if it holds true.

First, we can rewrite the expression for f as f = ± 4√(d/e) = ± 4√(d) / √(e).

Now, substitute this expression for f into the original equation:

d = 16e(± 4√(d) / √(e))^2

Expand the squared term:

d = 16e(± 16d / e)

Simplify the expression:

d = ± 256d

This equation is not true for all values of d, so option A is not a valid solution.

Option B: f = ± 4√(de) / e

Next, let's examine option B: f = ± 4√(de) / e. We can substitute this expression for f into the original equation and see if it holds true.

First, we can rewrite the expression for f as f = ± 4√(de) / e = ± 4√(d) / √(e).

Now, substitute this expression for f into the original equation:

d = 16e(± 4√(d) / √(e))^2

Expand the squared term:

d = 16e(± 16d / e)

Simplify the expression:

d = ± 256d

This equation is not true for all values of d, so option B is not a valid solution.

Option C: f = ± √(d/16e)

Finally, let's examine option C: f = ± √(d/16e). We can substitute this expression for f into the original equation and see if it holds true.

First, we can rewrite the expression for f as f = ± √(d/16e) = ± √(d) / √(16e).

Now, substitute this expression for f into the original equation:

d = 16e(± √(d) / √(16e))^2

Expand the squared term:

d = 16e(± d / 16e)

Simplify the expression:

d = ± d

This equation is true for all values of d, so option C is a valid solution.

Conclusion

In conclusion, the correct solution for f in the equation d = 16ef^2 is f = ± √(d/16e). This solution is verified by substituting the expression for f into the original equation and simplifying the resulting expression.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

Start with the given equation: d = 16ef^2
Divide both sides of the equation by 16e: d/16e = f^2
Take the square root of both sides of the equation: √(d/16e) = f
Simplify the expression: f = ± √(d/16e)

Final Answer

The final answer is f = ± √(d/16e).
Solving for f in the Equation d = 16ef^2: Q&A

Q: What is the given equation?

A: The given equation is d = 16ef^2, where d, e, and f are variables.

Q: What is the goal of the problem?

A: The goal of the problem is to solve for f in the given equation.

Q: How do we start solving for f?

A: We start by dividing both sides of the equation by 16e, which gives us d/16e = f^2.

Q: What is the next step in solving for f?

A: The next step is to take the square root of both sides of the equation, which gives us √(d/16e) = f.

Q: Why do we take the square root of both sides of the equation?

A: We take the square root of both sides of the equation because the square root is the inverse operation of squaring. This allows us to isolate f and solve for it.

Q: What is the final expression for f?

A: The final expression for f is f = ± √(d/16e).

Q: Why is the expression for f in the form of ± √(d/16e)?

A: The expression for f is in the form of ± √(d/16e) because the square root of a number can be positive or negative. This is represented by the ± symbol.

Q: What is the significance of the 16e term in the expression for f?

A: The 16e term in the expression for f is the denominator of the fraction under the square root. This term is important because it affects the value of f.

Q: Can we simplify the expression for f further?

A: Yes, we can simplify the expression for f further by canceling out the 16e term. This gives us f = ± √(d) / √(16e).

Q: What is the final answer to the problem?

A: The final answer to the problem is f = ± √(d/16e).

Common Mistakes to Avoid

Not dividing both sides of the equation by 16e: This is a crucial step in solving for f.
Not taking the square root of both sides of the equation: This is also a crucial step in solving for f.
Not simplifying the expression for f: Simplifying the expression for f can help to make it easier to understand and work with.

Tips and Tricks

Use algebraic manipulations to simplify the equation: Algebraic manipulations can help to make the equation easier to understand and work with.
Use the inverse operations to isolate f: The inverse operations of squaring and taking the square root can help to isolate f and solve for it.
Be careful with the ± symbol: The ± symbol represents the possibility of a positive or negative value. Be careful when working with the ± symbol to avoid making mistakes.

Conclusion

In conclusion, solving for f in the equation d = 16ef^2 involves using algebraic manipulations to simplify the equation and isolate f. By following the steps outlined in this article, you can solve for f and understand the significance of the 16e term in the expression for f.