Problem 52 - Brief 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 solutions are \(x = 2i\) and \(x = -2i\).

Step 1: Set the Equation to Zero

Start by moving the constant term to the other side of the equation. The equation is already set to zero: \[5x^2 + 20 = 0\]

Step 2: Simplify the Equation

Simplify the equation by dividing all the terms by the greatest common factor (GCF), which in this case is 5, to make the numbers more manageable:\[x^2 + 4 = 0\]

Step 3: Rearrange to Solve for x^2

Rearrange the equation to solve for \(x^2\) by moving the constant term to the other side of the equation:\[x^2 = -4\]

Step 4: Apply the Square Root

Take the square root of both sides to solve for \(x\). Remember that when you take the square root, you must consider both the positive and negative roots:\[x = \pm \sqrt{-4}\]

Step 5: Simplify using Imaginary Numbers

Since the square root of a negative number involves imaginary numbers, simplify using \(i\), where \(i = \sqrt{-1}\):\[x = \pm 2i\]

Step 6: State the Solution

The solutions to the equation are imaginary numbers:\(x = 2i\) and \(x = -2i\).

Key Concepts

Factoring Quadratics

Factoring a quadratic equation involves expressing it as a product of two binomials. This method is particularly useful when dealing with simple quadratics that can be easily factored to find the variable's roots. A quadratic equation generally takes the form \(ax^2 + bx + c = 0\).
To factor such equations, we look for two numbers that multiply to give \(ac\) and add up to \(b\). However, not all quadratic equations can be factored easily, especially when dealing with complex numbers or when the quadratic does not neatly split into real-number factors. In our example above, the equation \(5x^2 + 20 = 0\) simplifies to \(x^2 + 4 = 0\) after dividing by the GCF. This isn't factored easily because it leads to imaginary solutions when solving for \(x\).
Here, the factoring method wouldn't directly provide a solution, and we turn to other methods like completing the square or using the Quadratic Formula when dealing with more complex cases including imaginary components.

Quadratic Formula

The Quadratic Formula provides a foolproof method for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]This formula accounts for all possible scenarios including real, repeated, or imaginary solutions.
In our scenario, although our equation simplifies to \(x^2 = -4\), using the Quadratic Formula could still be helpful if we had non-zero linear terms. After simplification, substituting the values \(a = 1\), \(b = 0\), and \(c = 4\) into the formula confirms our imaginary solutions. Since the result under the square root (the discriminant) is negative, it indicates that the solutions are imaginary numbers. The quadratic formula seamlessly bridges over to help especially when other methods like factoring are not applicable or convenient.

Imaginary Numbers

Imaginary numbers can initially seem confusing; however, they are simply an extension of the regular real number system. An imaginary number involves \(i\), which is defined as \(\sqrt{-1}\).
This allows us to perform mathematical operations with square roots of negative numbers, essentially providing solutions to equations that have no real number solutions.
In the step-by-step solution of the quadratic \(x^2 = -4\), the square root results in \(\pm \sqrt{-4}\). Using imaginary numbers, this simplifies to \(\pm 2i\).

• "\(i\)" acts as a unit of measurement for imaginary numbers, similar to how 1 acts within the real numbers.
• Imaginary numbers are often represented on a plane perpendicular to the real number line in complex number charts, creating a comprehensive system that is vital in advanced fields such as engineering and physics.
• Understanding imaginary numbers allows for full solution sets to quadratics like our task, expanding the mathematical toolkit beyond real numbers alone.