Problem 52 - Applied Calculus

Question

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ 5 x^{2}+20=0 $$

Step-by-Step Solution

Answer

The equation has no real solutions; the solutions are complex.

Step 1: Set the Equation to Standard Form

The given equation is \( 5x^2 + 20 = 0 \). Start by setting the equation to standard quadratic form \( ax^2 + bx + c = 0 \). Here, it is already in standard form, where \( a = 5 \), \( b = 0 \), and \( c = 20 \).

Step 2: Solve by Quadratic Formula

Since the equation does not factor easily, use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute \( a = 5 \), \( b = 0 \), and \( c = 20 \) into the formula:\[ x = \frac{-0 \pm \sqrt{0^2 - 4 \cdot 5 \cdot 20}}{2 \cdot 5} \]

Step 3: Calculate the Discriminant

Calculate the discriminant, which is \( b^2 - 4ac \):\[ 0^2 - 4 \cdot 5 \cdot 20 = -400 \]Since the discriminant is negative, the equation has no real solutions.

Step 4: Interpret the Discriminant Result

As the discriminant is negative, the equation has complex solutions. This means the solutions fall into the realm of complex numbers, indicating that there are no x-intercepts on the real number line.

Key Concepts

Quadratic Formula

In mathematics, a key tool for finding the solutions to quadratic equations is the quadratic formula. Quadratic equations take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The quadratic formula provides a way to find the values of the variable \( x \) that satisfy the equation. The formula is as follows:

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

This formula is very useful because:

• It can solve any quadratic equation, regardless of whether it can be easily factored.
• The "\( \pm \)" symbol indicates two possible solutions, corresponding to two roots of the equation.

Using the quadratic formula involves plugging the coefficients from the equation into the formula and simplifying. Remember, the expression under the square root, \( b^2 - 4ac \), is known as the discriminant, which plays a crucial role in determining the nature of the roots.

Complex Numbers

Complex numbers extend the idea of numbers beyond the "real" numbers you may already know, like integers, fractions, or decimals. A complex number consists of a real part and an imaginary part, written as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.

• The imaginary unit \( i \) is defined such that \( i^2 = -1 \).

When the discriminant in the quadratic formula is negative, such as in the solution \( -400 \) for our exercise, the equation has no real solutions. Instead, the solutions are complex numbers.

• To find complex solutions using the quadratic formula, including negative discriminants, means the solution will be in terms of \( i \).

Complex numbers are especially important in higher mathematics, physics, and engineering, providing meaningful ways to calculate and interpret solutions that aren't "visible" on the real number line.

Discriminant

The discriminant of a quadratic equation is a key value found using the formula \( b^2 - 4ac \). Its value provides important information about the nature of the roots of the equation without needing to solve it fully. Here’s what the discriminant tells you:

• If \( b^2 - 4ac > 0 \), there are two distinct real solutions.
• If \( b^2 - 4ac = 0 \), there is exactly one real solution, often called a repeated or double root.
• If \( b^2 - 4ac < 0 \), there are no real solutions. Instead, the solutions will be complex numbers.

For example, in the given exercise, the discriminant \(-400\) indicates that the equation \(5x^2 + 20 = 0\) has complex solutions. Understanding the discriminant is essential because it immediately gives insight into the type of roots without solving the entire equation. This quick assessment helps decide the method forward for solving quadratics.