Problem 99 - Elementary Algebra

Question

Solve. $$ y 2-y+5=0 $$

Step-by-Step Solution

Answer

This quadratic has two complex solutions: \(y = \frac{1 \pm i\sqrt{19}}{2}\).

Step 1: Rewrite the Equation

First, let's recognize this is a quadratic equation in a slightly incorrect formatting. Rewrite the equation in the standard quadratic form: \[y^2 - y + 5 = 0\] This helps us clearly see the terms and identify them: \(a = 1\), \(b = -1\), and \(c = 5\).

Step 2: Identify the Discriminant

The discriminant \(D\) for a quadratic equation \(ax^2 + bx + c = 0\) is calculated as \(b^2 - 4ac\). Substituting the values, we find: \[D = (-1)^2 - 4 \times 1 \times 5 = 1 - 20 = -19\]

Step 3: Interpret the Discriminant

Since the discriminant \(D = -19\) is less than zero, this indicates that the quadratic equation has no real solutions. Instead, it has two complex conjugate solutions.

Step 4: Solve Using the Quadratic Formula

Since \(D < 0\), use the quadratic formula to find the complex roots:\[y = \frac{-b \pm \sqrt{D}}{2a}\]Plugging in the values, we get:\[y = \frac{-(-1) \pm \sqrt{-19}}{2 \times 1} = \frac{1 \pm i\sqrt{19}}{2}\]Thus the solutions are:\[y = \frac{1 + i\sqrt{19}}{2}\] and \[y = \frac{1 - i\sqrt{19}}{2}\]

Key Concepts

Discriminant

The discriminant is a crucial component of quadratic equations. It helps us determine the nature of the roots without actually solving the equation. For any quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant \(D\) is calculated as \(D = b^2 - 4ac\).
Here's how it affects the solutions:

• If \(D > 0\), the equation has two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, due to the roots being repeated.
• If \(D < 0\), as in our example \(D = -19\), the equation has no real solutions, but two complex conjugate solutions.

By calculating the discriminant first, you can anticipate the nature of the solutions, which is particularly helpful when solving or interpreting quadratic equations. In this problem, the negative discriminant means that we will be delving into the realm of complex numbers.

Complex Solutions

Complex numbers come into play when dealing with quadratic equations whose discriminant is negative. Such an equation does not intersect the x-axis and therefore lacks real roots. Instead, solutions arise from the concept of complex conjugates.
Complex numbers are expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined by the property \(i^2 = -1\).
In this exercise, as calculated, we found the roots to be \(y = \frac{1 + i\sqrt{19}}{2}\) and \(y = \frac{1 - i\sqrt{19}}{2}\).

• These are called complex conjugates because they have the same real part but opposite imaginary parts.
• They visually represent solutions that are symmetric about the real axis in the complex plane.

Understanding these solutions is essential for fully grasping how quadratic equations behave with negative discriminants.

Quadratic Formula

The quadratic formula is a powerful tool used to solve any quadratic equation of the form \(ax^2 + bx + c = 0\). It is given by:\[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Importantly, this formula systematically gives us the two possible values of \(y\), known as the roots, that satisfy the equation.
With a negative discriminant, as in our problem, the solutions transliterate into complex numbers. Plugging the values \(a = 1\), \(b = -1\), and \(c = 5\), into the formula reveals:

• The numerator \(-b\) becomes \(1\) as \(-(-1) = 1\).
• The discriminant \(\sqrt{-19}\) involves \(i\) for the imaginary part.
• Dividing by \(2a\), or \(2 \times 1\), simplifies the denominator to \(2\).

This precise calculation mechanism allows us to clearly derive: \(y = \frac{1 \pm i\sqrt{19}}{2}\).
Ultimately, the quadratic formula not only simplifies the process of finding roots but also reveals whether solutions are real or complex by the discriminant within.