Problem 67 - Elementary Algebra

Question

Solve using the quadratic formula. $$ -y 2-y-2=0 $$

Step-by-Step Solution

Answer

The solutions are \(y = -\frac{1}{2} + \frac{i\sqrt{7}}{2}\) and \(y = -\frac{1}{2} - \frac{i\sqrt{7}}{2}\).

Step 1: Write the Quadratic Equation in Standard Form

The first step is to write the given equation in the standard form of a quadratic equation, which is in the form \(ax^2 + bx + c = 0\). The given equation is \(-y^2 - y - 2 = 0\). It is already in the standard form where \(a = -1\), \(b = -1\), and \(c = -2\).

Step 2: Identify a, b, and c

Identify the coefficients from the quadratic equation. Here, \(a = -1\), \(b = -1\), and \(c = -2\). These values will be used in the quadratic formula.

Step 3: Apply the Quadratic Formula

The quadratic formula is \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = -1\), \(b = -1\), and \(c = -2\) into this formula.

Step 4: Calculate the Discriminant

The discriminant is given by \(b^2 - 4ac\). Substitute the values to get \((-1)^2 - 4(-1)(-2) = 1 - 8 = -7\). Since the discriminant is negative, it indicates that the solutions are complex numbers.

Step 5: Calculate the Complex Roots

Since the discriminant \(-7\) is negative, find the roots by substituting back into the quadratic formula. We get: \(y = \frac{-(-1) \pm \sqrt{-7}}{2(-1)} = \frac{1 \pm i\sqrt{7}}{-2}\). Simplified, this results in: \(y = -\frac{1}{2} \pm \frac{i\sqrt{7}}{2}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, providing an exact solution for any quadratic equation of the form \(ax^2 + bx + c = 0\). It is given by:

• \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In this formula, \(a\), \(b\), and \(c\) are the coefficients of the equation. The symbol \(\pm\) suggests there will be two solutions for y. This formula helps to solve any quadratic equation, even when it’s difficult or impossible to factor. Using the quadratic formula involves three main steps: plugging in the values of \(a\), \(b\), and \(c\), calculating the discriminant, and then finding the solutions.

Complex Solutions

When solving quadratic equations using the quadratic formula, you might encounter solutions that are not real numbers. These are called complex solutions. Complex numbers arise when the discriminant is negative, making it impossible to find real square roots.

In such cases, solutions involve imaginary numbers, represented by \(i\), where \(i = \sqrt{-1}\). Complex solutions can be written as a combination of a real part and an imaginary part. For example, if the solutions involve \(\pm \sqrt{-7}\), the answer is given as:

• \(y = \frac{1 \pm i\sqrt{7}}{-2}\)
This means that the complex solutions are \(-\frac{1}{2} + \frac{i\sqrt{7}}{2}\) and \(-\frac{1}{2} - \frac{i\sqrt{7}}{2}\). Understanding complex solutions is important as it broadens our capability to solve all kinds of quadratic equations.

Discriminant

The discriminant is a crucial part of solving quadratic equations using the quadratic formula. It is expressed as \(b^2 - 4ac\) and helps determine the nature of the solutions.

• If the discriminant is positive, the quadratic equation has two distinct real solutions.
• If it is zero, there is exactly one real solution, or two identical real roots.
• If negative, as in our example where it was \(-7\), the equation has two complex solutions.

The discriminant not only serves as a diagnostic tool but also simplifies the process of solving and understanding quadratic equations by providing insights into the type of roots you can expect. Knowing how to work with the discriminant is essential in analyzing any quadratic equation effectively.