Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Match each equation in Column I with its solution \((s)\) in Column \(I I\). A. \(\pm 2 i\) B. \(\pm 2 \sqrt{2}\) C. \(\pm i \sqrt{2}\) D. 2 E. \(\pm \sqrt{2} \quad\) F. \(-2\) G. \(\pm 2\) H. \(\pm 2 i \sqrt{2}\) $$x^{2}+2=0$$

Step-by-Step Solution

Answer

The solution is C: \(\pm i \sqrt{2}\).

Step 1: Start with the given equation

The given equation is \(x^2 + 2 = 0\). Our objective is to solve for \(x\).

Step 2: Isolate the quadratic term

To isolate the quadratic term \(x^2\), subtract 2 from both sides of the equation: \(x^2 = -2\).

Step 3: Solve for x using square roots

To solve for \(x\), take the square root of both sides, remembering to include both positive and negative roots. This gives \(x = \pm \sqrt{-2}\).

Step 4: Simplify the square root

Notice that \(-2\) can be expressed as \(-1 \times 2\). Thus, \(\sqrt{-2} = \sqrt{-1} \times \sqrt{2}\). The square root of \(-1\) is \(i\), so \(x = \pm i \sqrt{2}\).

Step 5: Match with the given solutions

Match the solution \(x = \pm i \sqrt{2}\) with its corresponding option in Column II. The matching solution is option C: \(\pm i \sqrt{2}\).

Key Concepts

Quadratic Equations

Quadratic equations are a foundational element in algebra that involve a variable with a degree of two. They are typically written in the standard form:

• \[ ax^2 + bx + c = 0 \]

where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The goal when solving these equations is to find the values of the variable \(x\) that satisfy the equation. Quadratic equations can have different types of solutions based on the discriminant \(b^2 - 4ac\). It determines whether the solutions are:

• Real and distinct
• Real and equal
• Complex (non-real)

In cases where the quadratic equation results in negative discriminants, the solutions involve complex numbers. Quadratic equations can be solved by various methods, such as factoring, completing the square, or using the quadratic formula. But when the equation cannot be easily factored, as in our exercise problem \(x^2 + 2 = 0\), we often resort to using square root operations.

Imaginary Unit

The imaginary unit \(i\) is a critical concept when dealing with square roots of negative numbers. It is defined by the property that \(i^2 = -1\). This definition allows us to work with negative square roots. Whenever we encounter \(\sqrt{-a}\), we can express it in terms of \(i\) as \(i\sqrt{a}\).Using \(i\) enables mathematicians to extend number operations to include complex numbers, which have the form

• \( a + bi \)

where \(a\) and \(b\) are real numbers. In the context of our problem, we found \(x = \pm i \sqrt{2}\) because

• \( \sqrt{-2} = \sqrt{-1 \times 2} = \sqrt{-1} \times \sqrt{2} = i\sqrt{2} \).

Understanding \(i\) is essential for solving equations that include negative square roots, like those derived from quadratic equations with negative discriminants.

Square Roots

Square roots are operations used to find a number that, when multiplied by itself, gives the original number. When solving quadratic equations, taking the square root of both sides can help isolate the variable. However, when dealing with negative square roots, such as \(\sqrt{-2}\), it introduces the use of the imaginary unit \(i\).In the step-by-step solution of the problem \(x^2 + 2 = 0\), we isolated \(x^2\) leading to

• \(x^2 = -2\).

To find \(x\), we took the square root of both sides:

• \(x = \pm \sqrt{-2} = \pm i \sqrt{2}\).

Always remember to consider both the positive and negative roots due to the property of squaring being reversible by either side in quadratic equations. This dual nature of square roots is what gives us two possible solutions for \(x\). This is crucial in matching solutions to non-linear equations.

Equation Solving

Equation solving is the process of finding unknown variables that satisfy the given equations. For quadratic equations, the goal is to determine the values of \(x\) that make the equation true. Solving involves isolating the variable \(x\) and simplifying, often requiring knowledge of operations with imaginary numbers and square roots.For the equation \(x^2 + 2 = 0\), we followed these general steps:

• First, rearranged to isolate the quadratic term: \(x^2 = -2\).
• Then, solved for \(x\) by taking the square root of both sides, considering both \( \pm \sqrt{-2} \).
• Finally, simplified using the imaginary unit: \(x = \pm i \sqrt{2}\).

With these steps, we concluded the solutions fit option C in the original exercise. Learning to solve such equations requires practice and familiarity with algebraic manipulation, the properties of imaginary numbers, and recognizing patterns in square roots. Mastery of these concepts is vital for deeper exploration into algebra and beyond.