Problem 40 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation by using the quadratic formula. $$ 4 x^{4}-21 x^{2}+5=0 $$

Step-by-Step Solution

Answer

We can solve the equation \(4x^4 - 21x^2 + 5 = 0\) using substitution and the quadratic formula. By letting \(y = x^2\), the equation becomes \(4y^2 - 21y + 5 = 0\). Solving for \(y\) using the quadratic formula gives \(y_1 = 5\) and \(y_2 = \frac{1}{4}\). By substituting back the values of \(y\), we find the possible values of \(x\): \(x = \pm\sqrt{5}, \pm\frac{1}{2}\).

Step 1: Make a substitution

Let \(y = x^2\), then the given equation becomes a quadratic equation in terms of \(y\): $$ 4y^2 - 21y + 5 = 0 $$

Step 2: Use the quadratic formula to solve for \(y\)

Use the quadratic formula: \(y = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), where \(a = 4\), \(b = -21\), and \(c = 5\). Plug in the values: $$ y = \frac{-(-21) \pm \sqrt{(-21)^2 - 4 (4)(5)}}{2(4)} $$ Simplify the equation and solve for \(y\): $$ y = \frac{21 \pm \sqrt{441 - 80}}{8} $$ $$ y = \frac{21 \pm \sqrt{361}}{8} $$ $$ y = \frac{21 \pm 19}{8} $$ Now we have two possible values for \(y\): $$ y_1 = \frac{21 + 19}{8} = 5 $$ $$ y_2 = \frac{21 - 19}{8} = \frac{1}{4} $$

Step 3: Solve for \(x\)

Now that we have found the values of \(y\), we need to find the corresponding values of \(x\) by using the fact that \(y = x^2\). For \(y_1 = 5\): $$ x^2 = 5 $$ $$ x = \pm\sqrt{5} $$ For \(y_2 = \frac{1}{4}\): $$ x^2 = \frac{1}{4} $$ $$ x = \pm\frac{1}{2} $$

Step 4: State the final solutions

We've found all the possible values for \(x\): $$ x = \pm\sqrt{5}, \pm\frac{1}{2} $$

Key Concepts

Polynomial Equation

A polynomial equation is an expression composed of variables and coefficients, involving terms with integer exponents. These equations can range from the simple linear form to complex ones like quartic, which we are dealing with here. In the given exercise, the polynomial equation is:\[ 4x^4 - 21x^2 + 5 = 0 \]Here, the highest degree of the variable \(x\) is 4, indicating a quartic polynomial equation. The goal is to find the values of \(x\) such that the equation holds true. Solving these kinds of equations often involves techniques like factoring, completing the square, or using the quadratic formula if they can be transformed into a quadratic form. Recognizing the degrees of polynomial terms is crucial, as they give us insight into the fundamental nature of the equation and guide us towards choosing suitable solution methods.

Substitution Method

The substitution method is a powerful tool in solving higher-degree polynomial equations by simplifying them. The idea is to reduce a complex equation into a more manageable one by introducing a new variable. In this exercise, the original equation involves \(x^4\), which can be tricky to work with directly.By introducing substitution, we let \(y = x^2\). Through this substitution, our original equation simplifies to:\[ 4y^2 - 21y + 5 = 0 \]This conversion effectively changes our quartic equation into a quadratic one, making it easier to solve using techniques familiar to quadratic equations. The substitution method not only simplifies the solving process but also provides a systematic approach to restructuring mathematical problems which might be intimidating in their original forms.

Mathematical Solution Steps

Mathematical solutions often involve several steps to arrive at the final answer. Each step needs to be understood for a clear grasp of how solutions are derived. In solving the equation using the quadratic formula, we proceed with precise steps:

• First, we apply our substitution: \(y = x^2\), rewriting the equation as \(4y^2 - 21y + 5 = 0\).
• Next, we recognize the quadratic form and apply the quadratic formula: \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), identifying \(a = 4\), \(b = -21\), and \(c = 5\).
• After substituting these values into the formula, we calculate the discriminant: \(b^2 - 4ac = 441 - 80 = 361\).
• Using the discriminant, solve for \(y\): \(y = \frac{21 \pm 19}{8}\) yields \(y_1 = 5\) and \(y_2 = \frac{1}{4}\).
• Finally, revert back to the original variable \(x\) by solving \(x^2 = y\). For \(y_1 = 5\), \(x = \pm \sqrt{5}\), and for \(y_2 = \frac{1}{4}\), \(x = \pm \frac{1}{2}\).

Each of these steps is crucial and ensures a complete understanding of the path from the problem to the solution. By working through each calculation patiently, the mathematical journey becomes clearer and more rewarding.