Problem 43 - College Algebra with Corequisite Support

Question

For the following exercises, solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution. $$ 4+\frac{1}{x}-\frac{1}{x^{2}}=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{-1 + \sqrt{17}}{8}\) and \(x = \frac{-1 - \sqrt{17}}{8}\).

Step 1: Clear Fractions

Let's start by eliminating the fractions. Multiply through by the common denominator, which is \(x^2\), to clear all fractions: \(x^2 \left(4 + \frac{1}{x} - \frac{1}{x^2}\right) = 0\). This simplifies to \(4x^2 + x - 1 = 0\).

Step 2: Identify Quadratic Coefficients

Now that we have \(4x^2 + x - 1 = 0\), identify the coefficients: \(a = 4\), \(b = 1\), and \(c = -1\).

Step 3: Use the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the coefficients into the formula: \(x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 4 \cdot (-1)}}{2 \cdot 4}\).

Step 4: Simplify Under the Square Root

Calculate the discriminant: \(1^2 - 4 \times 4 \times (-1) = 1 + 16 = 17\). Therefore, \(x = \frac{-1 \pm \sqrt{17}}{8}\).

Step 5: Write the Solution

The solutions of the quadratic equation are \(x = \frac{-1 + \sqrt{17}}{8}\) and \(x = \frac{-1 - \sqrt{17}}{8}\). These solutions are real.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of quadratic equations. A quadratic equation is generally written in the form of: \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The quadratic formula can be expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula allows us to find the roots of the equation, which are the values of \( x \) that make the equation true.

• \(-b\) indicates that the coefficient \( b \) is negated.
• \(\pm\) means there could be two possible solutions.
• \(\sqrt{b^2 - 4ac}\) is the square root of the discriminant.

Using the quadratic formula is systematic and reliable, alleviating guesswork and providing solutions when factoring isn’t straightforward.

Discriminant

Understanding the discriminant is essential when solving quadratic equations. The discriminant is found within the quadratic formula and is represented as: \[ b^2 - 4ac \] This value provides critical information about the nature of the equation’s roots. Here's how the discriminant determines the nature of the roots:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real solutions.
• If \( b^2 - 4ac = 0 \), there is one real solution, also known as a double root.
• If \( b^2 - 4ac < 0 \), the solutions are complex or imaginary, meaning there are no real roots.

In the exercise, the discriminant was found to be 17, a positive number, indicating two distinct real solutions. Identifying the discriminant will quickly give you insights into what kinds of solutions to expect.

Solving Quadratics

To solve a quadratic equation using the quadratic formula, you need to follow a step-by-step approach. Firstly, ensure the equation is in the correct quadratic form \( ax^2 + bx + c = 0 \). If the equation involves fractions, clear them so that the quadratic form becomes evident, as shown in the exercise by multiplying through by the common denominator \( x^2 \).

• Identify the coefficients \(a\), \(b\), and \(c\).
• Use these coefficients in the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions.
• Simplify any terms within the square root to determine the roots.

It's crucial to simplify each part of the calculation as much as possible and to be careful with signs, especially since \(-b\) and the term under the square root can significantly change your answers. Finally, make sure to check if your solutions are real by referring to the value of the discriminant.