Problem 38 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation by using the quadratic formula. $$ m^{4}-13 m^{2}+36=0 $$

Step-by-Step Solution

Answer

The solutions to the given equation \(m^4 - 13m^2 + 36 = 0\) are \(m = -3, 3, -2\), and \(2\).

Step 1: Identify the coefficients of the quadratic equation

Given the equation \(m^4 - 13m^2 + 36 = 0\). This is a quadratic equation in the variable \(m^2\). We can identify the coefficients as follows: \(a = 1\) \(b = -13\) \(c = 36\)

Step 2: Apply the quadratic formula

The quadratic formula states that for any quadratic equation of the form \(ax^2 + bx + c = 0\), the solutions can be found using the following formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] In this case, we are solving for \(m^2\), so: \[m^2 = \frac{-(-13) \pm \sqrt{(-13)^2 - 4(1)(36)}}{2(1)}\]

Step 3: Simplify the expression

Now let's simplify the expression: \[m^2 = \frac{13 \pm \sqrt{169 - 144}}{2}\] \[m^2 = \frac{13 \pm \sqrt{25}}{2}\] \[m^2 = \frac{13 \pm 5}{2}\] This gives us two possible solutions for the variable \(m^2\): 1. \(m^2 = \frac{13 + 5}{2} = \frac{18}{2} = 9\) 2. \(m^2 = \frac{13 - 5}{2} = \frac{8}{2} = 4\)

Step 4: Find the values of m

Now we need to find the values of m by taking the square root of the solutions for \(m^2\): 1. For \(m^2 = 9\): \(m = \pm(\sqrt{9}) = \pm3\) 2. For \(m^2 = 4\): \(m = \pm(\sqrt{4}) = \pm2\) Hence, the solutions to the given equation are \(m = -3, 3, -2\), and \(2\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation characterized by the highest exponent of the variable being 2. It's usually presented in the form \(ax^2 + bx + c = 0\). In this problem, we have \(m^4 - 13m^2 + 36 = 0\), which might seem unusual at first. However, by substituting \(x = m^2\), it becomes a familiar quadratic equation in terms of \(x\). This allows us to use the well-known quadratic strategies to solve it. Quadratic equations are fundamentally vital because they appear in various real-world scenarios from physics to finance.

Coefficients

In quadratic equations, coefficients are the numerical parts that multiply the variables. For the equation \(ax^2 + bx + c = 0\), \(a\), \(b\), and \(c\) are the coefficients. They help define the shape and position of the parabola on a graph. In our example, the coefficients are:

• \(a = 1\), which is associated with the \(m^4\) term, but when transformed using \(x = m^2\), it becomes the leading coefficient of \(x^2\).
• \(b = -13\), which modifies the linear term to \(m^2\).
• \(c = 36\), which is the constant term affecting the equation’s intercept.

Understanding coefficients is essential because they directly impact the roots of the equation.

Square Root

The square root is a number that, when multiplied by itself, yields the original number. In solving quadratic equations, the square root appears in the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this problem, after simplifying within the square root, we have \(b^2 - 4ac = 25\). Thus, \(\sqrt{25} = 5\). The square root is crucial because it reveals the principal parts of the roots of quadratic equations. Without it, we would miss essential solutions applicable to the equation.

Solving Equations

Solving equations involves finding the values of variables that make the equation true. For quadratic equations, this usually means finding the values of the variable \(x\) (or \(m\) in our transformed equation). Here’s a quick step-through of the process:

• Identify coefficients \(a\), \(b\), and \(c\).
• Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Simplify to find the roots, such as \(m^2 = 9\) or \(m^2 = 4\).
• Finally, take the square root to find \(m\), leading to solutions \(m = -3, 3, -2, 2\).

This structured approach ensures you quickly find accurate solutions to quadratic equations.