Problem 39 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The solutions are \(x = \frac{5}{2}\) and \(x = -\frac{5}{2}\).

Step 1: Isolate the Power Term

We start with the equation \(16x^4 = 625\). We want to isolate \(x^4\) by dividing both sides by 16:\[x^4 = \frac{625}{16}\]

Step 2: Take the Fourth Root

To solve for \(x\), take the fourth root of both sides. Recall that the fourth root of a number is equivalent to raising the number to the power of \(\frac{1}{4}\):\[x = \pm\left(\frac{625}{16}\right)^{\frac{1}{4}}\]

Step 3: Simplify the Expression

Now simplify \(\left(\frac{625}{16}\right)^{\frac{1}{4}}\). We know \(625 = 25^2\) and \(16 = 4^2\), so:\[x = \pm\left(\frac{25^2}{4^2}\right)^{\frac{1}{4}} = \pm\left(\frac{25}{4}\right)^{\frac{1}{2}}\]Taking the square root gives us:\[x = \pm\frac{5}{2}\]

Step 4: Graphical Verification

Graph the functions \(y = 16x^4\) and \(y = 625\) on the same set of axes. The solutions to the equation \(16x^4 = 625\) are the x-values where these curves intersect.Upon graphing, observe that the intersections occur at \(x = \frac{5}{2}\) and \(x = -\frac{5}{2}\), confirming our algebraic solutions.

Key Concepts

Algebraic Solution

Solving a polynomial equation algebraically involves manipulating the equation to isolate the desired variable and solve for its value. In the given exercise, we start with the equation:

• 16x^4 = 625.

The first step in finding an algebraic solution is to isolate the term with the variable, which in this case, is the term containing x. By dividing both sides of the equation by 16, we isolate x\(^4\):

• \(x^4 = \frac{625}{16}\).

The next step involves taking the fourth root of both sides to solve for x. Recall that taking the fourth root is equivalent to raising each side to the power of \(\frac{1}{4}\):

• \(x = \pm\left(\frac{625}{16}\right)^{\frac{1}{4}}\).

Simplifying the expression further involves recognizing that both 625 and 16 are perfect squares:

• 625 = 25\(^2\)
• 16 = 4\(^2\).

Thus, we can rewrite and simplify the expression as:

• \(x = \pm\left(\frac{25}{4}\right)^{\frac{1}{2}} = \pm\frac{5}{2}\).

Hence, the solutions are \(x = \frac{5}{2}\) and \(x = -\frac{5}{2}\).

Graphical Solution

To graphically solve a polynomial equation, we need to interpret the equation as a comparison of two function outputs. This means plotting each side of the equation as a separate function. For this particular exercise, we compare:

• \(y = 16x^4\)
• \(y = 625\).

By graphing both functions on the same axis, we can find the solutions by identifying the x-values where the two graphs intersect.
The function \(y = 16x^4\) is a curve that is symmetric with respect to the y-axis and strikes upwards, while \(y = 625\) is a horizontal line. The points where these two graphs meet represent our solutions.
Upon examining the graph, you'll see intersections at points \(x = \frac{5}{2}\) and \(x = -\frac{5}{2}\), verifying our algebraic results. This graphical approach provides a visual confirmation of the solutions, reinforcing the results acquired through algebraic methods.

Fourth Roots

Understanding fourth roots is central to solving equations like the one provided in this exercise. A fourth root of a given number is a value that, when raised to the power of four, yields the original number. Simply put, if \(x^4 = a\), then the fourth root of a is \(x = \pm a^{\frac{1}{4}}\).
Fourth roots often involve recognizing perfect squares, as seen in the equation \(16x^4 = 625\), where we found:

• \(\frac{625}{16}\) is simplified by identifying that
• 625 = 25\(^2\) and 16 = 4\(^2\).

This reduces our problem to finding the square root of a simpler fraction.
Calculating the fourth root efficiently involves breaking it into two steps:

• First, simplify the expression within the fraction, or identify factors (in this case 25 and 4),
• Then, find the square root of the resulting simplified fraction: \((25/4)^{1/2} = 5/2\).

Fourth roots therefore extend beyond basic arithmetic into recognizing patterns and simplifying.

Intersection of Functions

The concept of intersection is an insightful aspect of solving equations graphically. When you have two functions, their intersection points on a graph provide the solutions to the equation formed by equaling those two functions.
For the equation 16\(x^4 = 625\), consider both sides as individual functions:

• Function A: \(y = 16x^4\)
• Function B: \(y = 625\).

The intersection points occur where Function A's curve meets the horizontal line of Function B, meaning the y-values are the same for the same x-value.
Graphing reveals exactly these points where the equation's solutions reside, specifically \(x = \frac{5}{2}\) and \(x = -\frac{5}{2}\).
This demonstrates not only solutions but the approach of using graphical intersections to find variable solutions and gain a deeper understanding of how equations behave visually.