Problem 6 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 1 through 7, give exact answers, not numerical approximations. Solve: \((\pi x)^{2}=\pi x\). (There are two answers.)

Step-by-Step Solution

Answer

The two solutions to the equation \((\pi x)^{2}=\pi x\) are \(x = 0\) and \(x = 1\)

Step 1: Rewrite the original quadratic equation

First, make one side of given equation equal to zero. The quadratic equation should be in the form \(ax^{2} + bx + c = 0\). After rearranging, the new equation become \((\pi x)^2 - \pi x = 0\)

Step 2: Factor out

The next step is to factor out the common factor between \( (\pi x)^2\) and \(- \pi x\). Factoring out \(\pi x\) leave us to \(\pi x ( \pi x - 1) = 0\)

Step 3: Solve for x values

In order for the whole equation to be zero, one of the expressions in the multiplication must be zero. Setting \(\pi x = 0\) yield \(x = 0\) and solving \(\pi x - 1 = 0\) give \(x=1\). So, the two solutions for this equation are \(x = 0\) and \(x = 1\)

Key Concepts

Factoring Quadratics

When we encounter a quadratic equation, the method of 'factoring quadratics' is often the first strategy to consider. Quadratic equations are algebraic expressions presented in the form \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). The objective of factoring is to decompose the quadratic into a product of binomials or other expressions, making it easier to identify the solutions.
For instance, if we have a quadratic like \(x^2 - 5x + 6 = 0\), factoring this would involve finding two numbers that multiply to give the last term, 6, and add to give the middle coefficient, -5. In this case, the factors are \(x - 2\) and \(x - 3\). By factoring, we transform the equation into \(x - 2)(x - 3) = 0\), providing a simpler path to finding \(x\) values that satisfy the equation.

• Identify \(a\), \(b\), and \(c\) in the quadratic.
• Look for two numbers that multiply to \(ac\) and add to \(b\).
• Break the middle term into two terms using these numbers.
• Factor by grouping, where possible, to find binomial factors.

By properly factoring a quadratic, we often avoid more complex methods and quickly find the roots of the equation.

Quadratic Formula

For quadratics that are less straightforward to factor, the 'quadratic formula' is a powerful and universal tool. It provides the roots of any quadratic equation of the form \(ax^{2} + bx + c = 0\). The formula is derived from the process of completing the square and is stated as: \[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}.\]The symbol \(\pm\) indicates two solutions: one involving addition and the other involving subtraction. Here's how to use the quadratic formula in practical steps:

• Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation.
• Plug these values into the formula.
• Simplify under the square root, known as the discriminant.
• Determine the two possible values of \(x\) using the \(\pm\) operator.

Always ensure if the discriminant (\(b^2 - 4ac\)) is non-negative before proceeding, as this will determine the nature of the roots (real or complex) and their calculation.

Zero Product Property

The 'zero product property' is a vital algebraic principle stating that if a product of two factors equals zero, then at least one of the factors must be zero. It's expressed as \(ab = 0\) implying \(a = 0\) or \(b = 0\) or both. This property is especially useful when solving factored quadratic equations.
Consider the equation \( (x - 4)(x + 2) = 0\). Due to the zero product property, we deduce that \(x - 4 = 0\) or \(x + 2 = 0\). Solving these individually gives the roots \(x = 4\) or \(x = -2\), respectively. Here’s a general approach when applying the zero product property:

• Equations must be factored completely.
• Set each factor containing the variable equal to zero.
• Solve for the variable in each resulting equation.

This powerful property simplifies the process to find practical solutions once a quadratic is factored and serves as a cornerstone for many other algebraic procedures.

Algebraic Equations

Understanding 'algebraic equations' is fundamental for solving a broad spectrum of mathematical problems. An algebraic equation is an equality involving variables and constants that employ arithmetic operations like addition, subtraction, multiplication, and division. The goal when solving such equations is to isolate the variable on one side to find its value(s).
Through algebra, complex equations can be simplified using a variety of techniques, including factoring, the zero product property, and the quadratic formula mentioned before. These tools allow for transforming and simplifying equations to find exact values for the variables involved.

• Simplify the equation by combining like terms and removing parentheses where possible.
• Use the appropriate method (factoring, zero product property, substitution, elimination, etc.) based on the form of the equation.
• Check your solutions by substituting them back into the original equation to verify the equality holds.

By mastering these concepts and their applications, one can tackle and solve an extensive array of algebraic equations with confidence and precision.