Problem 33 - Algebra and Trigonometry

Question

Solve the equation both algebraically and graphically. $$ 16 x^{4}=625 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{25}{4} \) and \( x = -\frac{25}{4} \).

Step 1: Isolate the Variable Term

First, we want to isolate the term containing the variable. In this equation, the variable term is on one side marked as \(16x^4\). To do this, we divide both sides of the equation by 16.\[ x^4 = \frac{625}{16}\]

Step 2: Apply the Fourth Root

To solve for \(x\), we need to eliminate the exponent. We do this by taking the fourth root of both sides of the equation.\[ x = \pm \sqrt[4]{\frac{625}{16}}\]

Step 3: Simplify the Roots

Simplify the fourth root expression. Calculate the square root of both the numerator and the denominator of the fraction.\[ \sqrt{625} = 25 \quad \text{and} \quad \sqrt{16} = 4\]So, the equation becomes:\[ x = \pm \frac{25}{4}\]

Step 4: Graphical Verification

Now, let's cross-verify this solution graphically. Plot the two functions: \( y = 16x^4 \) and \( y = 625 \). The solutions where both functions intersect are the solution points of the equation. The graph should show intersections at: \( x = \frac{25}{4} \) and \( x = -\frac{25}{4} \).

Step 5: Conclusion

From both the algebraic step and the graphical observation, we conclude that the solutions to \(16x^4 = 625\) are \( x = \frac{25}{4} \) and \( x = -\frac{25}{4} \).

Key Concepts

Fourth Root

Understanding the concept of the fourth root is essential when dealing with equations that involve terms raised to the fourth power, like the one in our exercise. The fourth root of a number is a value that, when multiplied by itself four times, gives the original number. For instance, if you have \( x^4 = a \), to find \( x \), you have to take the fourth root of \( a \). This is similar to taking a square root, but instead, you are looking for a number that, when raised to the power of four, equals \( a \). In the equation \( x = \pm \sqrt[4]{\frac{625}{16}} \), finding the fourth root involves taking the square root twice for precision:

• First, find the square root of the numerator: \( \sqrt{625} = 25 \).
• Then, find the square root of the denominator: \( \sqrt{16} = 4 \).

Thus, the fourth root of \( \frac{625}{16} \) is the fraction \( \frac{25}{4} \). Remember that fourth roots can yield both positive and negative results, as multiplication by itself four times negates any sign.

Isolate Variable

Isolating the variable is a fundamental step in solving algebraic equations. The goal is to get the variable by itself on one side of the equation, which makes it easier to solve. When you start with an equation like \( 16x^4 = 625 \), you need to isolate \( x^4 \), the term containing the variable. You do this by performing inverse operations that keep the equation balanced. In this case, dividing both sides by 16 simplifies the equation to \( x^4 = \frac{625}{16} \). Once the variable term is isolated, you can proceed with solving the simplified equation. Keep in mind that maintaining the balance of the equation is key, so whatever operation you perform on one side, must be performed on the other side as well.

Graphical Verification

Graphical verification serves as a visual check for the solutions obtained algebraically. It ensures that the solutions make sense in a graphical context.In our example, you would plot

• \( y = 16x^4 \) on one graph, representing the left side of the original equation.
• \( y = 625 \) on another graph, representing the fixed value on the right side.

Where these two graphs intersect are your solution points. In the context of this problem, you should observe intersections at \( x = \frac{25}{4}\) and \( x = -\frac{25}{4} \). These points confirm the accuracy of the algebraic solutions, as they show where the two equations have equal values. This method is particularly helpful as it provides a visual intuition and confirmation of the algebraic method.

Solution Points

Solution points are the values of \( x \) that satisfy the equation, yielding equal expressions on both sides. They're the answers you derive from solving the equation. In an exercise like \( 16x^4 = 625 \), the solution points are specific values where the derived variable satisfies the original equation. From algebraic solutions, we find:

• \( x = \frac{25}{4} \)
• \( x = -\frac{25}{4} \)

These values are identified as solution points because you confirmed them both algebraically and graphically. In graphical terms, they represent where the plot of the function \( y = 16x^4 \) intersects with the line \( y = 625 \). These points illustrate the importance of verification in multiple forms to ensure both correctness and understanding of the solution.