Problem 18 - A Graphical Approach to College Algebra

Question

Find all complex solutions for each equation by hand. Do not use a calculator. $$1-\frac{3}{x}-\frac{10}{x^{2}}=0$$

Step-by-Step Solution

Answer

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

Step 1: Clear the Fractions

To eliminate the fractions in the equation, multiply through by the common denominator, which is \(x^2\). The equation becomes: \(x^2 imes 1 - x^2 imes \frac{3}{x} - x^2 imes \frac{10}{x^2} = 0\). Simplify this to get \(x^2 - 3x - 10 = 0\).

Step 2: Factor the Quadratic Equation

Look for two numbers that multiply to \(-10\) and add to \(-3\). These numbers are \(-5\) and \(2\). Therefore, the quadratic equation \(x^2 - 3x - 10 = 0\) can be factored as \((x - 5)(x + 2) = 0\).

Step 3: Solve for x

Using the zero product property, set each factor equal to zero: \(x - 5 = 0\) and \(x + 2 = 0\). Solving these gives \(x = 5\) and \(x = -2\).

Step 4: Verify the Solutions

Plug each solution back into the original equation to verify. For \(x = 5\), the original equation becomes \(1 - \frac{3}{5} - \frac{10}{25} = 0\), which simplifies to \(1 - \frac{3}{5} - \frac{2}{5} = 0\) which is true. For \(x = -2\), the equation becomes \(1 - \frac{3}{-2} - \frac{10}{4} = 0\), simplifying to \(1 + \frac{3}{2} - \frac{5}{2} = 0\) which is also true.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \). In its simplest form, this equation is a curve in the shape of a "U," known as a parabola, when plotted on a graph. Quadratic equations are fundamental in mathematics due to their wide applications in various fields such as physics, engineering, and finance. For example, they can describe the trajectory of an object in projectile motion or help determine the maximum profit in a business scenario.

The specific equation given in the exercise, \( x^2 - 3x - 10 = 0 \), is a standard quadratic equation. The task involves finding the values of \( x \) that satisfy this equation, known as its roots or solutions.

Factoring

Factoring is a method used to solve quadratic equations by expressing the quadratic polynomial as a product of linear factors. Essentially, it is about finding two binomials that multiply together to form the given quadratic equation. This is often possible when the quadratic can be easily broken down into simpler terms. The process involves looking for two numbers that multiply together to give the constant term (\(-10\) in our example) and add up to the coefficient of the \( x \) term (\(-3\) here).

In the exercise, the quadratic equation \( x^2 - 3x - 10 = 0 \) is factored into \((x - 5)(x + 2) = 0\). Here, \(-5\) and \(2\) are the numbers that work because \(-5\times 2 = -10\) and \(-5 + 2 = -3\). Factoring transforms the equation into a form that is easy to solve using the zero product property.

Zero Product Property

The zero product property is an important principle in algebra that states if the product of two factors is zero, then at least one of the factors must be zero. This is expressed mathematically as: if \( a \times b = 0 \), then \( a = 0 \) or \( b = 0 \).

Using this property provides a straightforward way to find the solutions of a factored quadratic equation. Once the quadratic equation \( x^2 - 3x - 10 = 0 \) is factored into \((x - 5)(x + 2) = 0\), we apply the zero product property. We set each factor individually equal to zero: \( x - 5 = 0 \) and \( x + 2 = 0 \). Solving these simple equations gives us the solutions \( x = 5 \) and \( x = -2 \). These values satisfy the original quadratic equation and are considered its roots. This method of solving is not only efficient but also highlights the mathematical beauty of quadratic equations.