Problem 12 - Algebra 2

Question

Solve each equation. $$ -5 x^{2}-25=0 $$

Step-by-Step Solution

Answer

The equation has no real solutions.

Step 1: Move Constant to One Side

Start with the equation: \[-5x^2 - 25 = 0\]Add 25 to both sides to isolate the term with the variable on one side:\[-5x^2 = 25\]

Step 2: Divide by Coefficient of \

The current equation is:\[-5x^2 = 25\]Divide every term by -5 to solve for \(x^2\):\[x^2 = -5\]

Step 3: Solve for x

Now, take the square root of both sides to solve for \(x\):\[x = \pm \sqrt{-5}\]Since you cannot take the square root of a negative number in the real number system, \(x\) is not a real number.

Key Concepts

Imaginary Numbers

Imaginary numbers can seem a bit tricky at first, but they are a fascinating and essential part of mathematics. Whenever we encounter the square root of a negative number, we enter the realm of imaginary numbers. The basic unit of imaginary numbers is denoted as \( i \), where \( i \) is defined as \( \sqrt{-1} \). Thus, when you see \( \sqrt{-5} \) in the solution, it means we can rewrite this using imaginary numbers, such as \( \sqrt{-5} = \sqrt{5}i \). This is because we express \( \sqrt{-1} \) as \( i \), making it possible to handle negative square roots without stepping outside number theory.

Here's how imaginary numbers work in solving equations:

• When you get a negative under a square root, think "imaginary" or "\( i \)".
• \( i^2 = -1 \), an identity that helps simplify complex expressions.
• Imaginary numbers help better understand complex, real-world systems such as electrical engineering and quantum physics.

Embrace imaginary numbers as an expansion of our number system, allowing us to solve equations that would otherwise have no solution in the realm of common real numbers.

Square Roots

The concept of square roots is one of the fundamental aspects of mathematics typically introduced early in algebra. A square root of a number \( a \) is defined as a number \( x \) such that \( x^2 = a \). However, the process becomes a bit more intricate when dealing with negative numbers. In the real number system, a negative under the square root implies no real solution, which introduces us to imaginary numbers.

To solve equations involving square roots, keep these points in mind:

• If \( a \) is positive, \( \sqrt{a} \) has two possible values, a positive \( x \) and a negative \( x \) (i.e., \( x = \pm \sqrt{a} \)).
• For negative \( a \), \( \sqrt{a} \) doesn't give a real number but an imaginary one.
• Understanding the implications of square roots means grasping their role in different number systems.

Mastery of square roots forms a strong foundation for tackling more advanced mathematical topics and problem-solving scenarios.

Real Number System

The real number system is a cornerstone of mathematics, encompassing a wide range of numbers from zero to infinity, including decimals and fractions. It includes both rational numbers (such as fractions and whole numbers) and irrational numbers (like \( \pi \) and \( \sqrt{2} \)). However, when you hit negative numbers under a square root, you step outside the realm of real numbers. This is where the imaginary number system comes into play.

Some important points about the real number system include:

• The real number line doesn’t have end points; it extends indefinitely in both directions.
• All real numbers can be expressed as decimals, although they might not always terminate (e.g., \( 1/3 = 0.333... \)).
• Real numbers are important for measuring and calculating in everyday life.

By understanding the boundaries of the real number system, you can better appreciate why imaginary numbers are not just useful but necessary for solving certain mathematical problems.