Problem 9 - Elementary Algebra

Question

Solve the following quadratic equations. \(x^{2}+20=0\)

Step-by-Step Solution

Answer

The solutions are \(x = \pm 2i\sqrt{5}\).

Step 1: Identify the quadratic equation

The given equation is \(x^2 + 20 = 0\). Identify the quadratic equation in the standard form \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = 0\), and \(c = 20\).

Step 2: Isolate the quadratic term

Subtract 20 from both sides of the equation to isolate the quadratic term: \(x^2 = -20\).

Step 3: Solve for x

Take the square root of both sides of the equation. Remember to include the \( \pm \) symbol as there are two solutions: \(x = \pm \sqrt{-20}\).

Step 4: Simplify the square root

Simplify the square root of -20. Recall that \( \sqrt{-1} = i\) where \(i\) is the imaginary unit. Thus, \(x = \pm i\sqrt{20} \).

Step 5: Further Simplify

Simplify \( \sqrt{20} \). Since \(20 = 4 \times 5\), \( \sqrt{20} \) can be written as \(2\sqrt{5}\). Therefore, \(x = \pm 2i\sqrt{5} \).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree 2, typically written in the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\). They are essential in algebra and appear frequently in various mathematical contexts. The solutions of quadratic equations can be found using several methods, such as factoring, completing the square, or applying the quadratic formula. Understanding how to solve quadratic equations is fundamental for further studies in mathematics and science.

Complex Numbers

Complex numbers extend the real number system to include solutions to equations like \(x^2 + 1 = 0\). In this case, we introduce the imaginary unit \(i\), where \(i^2 = -1\). Thus, complex numbers are written in the form \(a + bi\), where \(a\) and \(b\) are real numbers. To solve equations involving complex numbers, remember:

• The square root of a negative number involves \(i\).
• Operations with complex numbers follow similar rules as real numbers but include rules for \(i\).

Grasping complex numbers allows you to solve a wider range of quadratic equations and dive deeper into advanced mathematics.

Square Roots

Square roots involve finding a number which, when multiplied by itself, yields the original number. For example, the square root of 25 is \(5\) because \(5 \times 5 = 25\). When solving equations like \(x^2 + 20 = 0\), you encounter the square root of a negative number. This is where complex numbers come into play:

• \(\sqrt{-1} = i\)
• For negative numbers, \(\sqrt{-a} = i\sqrt{a}\).

Simplifying square roots often involves breaking down the number into factors and taking advantage of properties like \(\sqrt{ab} = \sqrt{a}\sqrt{b}\). With these skills, you can handle square roots in both real and complex contexts comfortably.