Problem 62 - Algebra and Trigonometry

Question

\(61-70\) . Find all solutions, real and complex, of the equation. $$ x^{4}-16=0 $$

Step-by-Step Solution

Answer

Real solutions: 2, -2. Complex solutions: 2i, -2i.

Step 1: Rearrange the Equation

Start by rearranging the equation to set it to zero: \[ x^4 - 16 = 0 \] The equation is already set to zero, so you can move to the next step.

Step 2: Factor the Equation

Recognize that the equation can be factored using the difference of squares. Recall the identity: \[ a^2 - b^2 = (a - b)(a + b) \]For our equation, \[ x^4 - 16 = (x^2)^2 - (4)^2 = (x^2 - 4)(x^2 + 4) \] Now we have: \[ (x^2 - 4)(x^2 + 4) = 0 \]

Step 3: Solve Each Factor Separately

Solve the equation by setting each factor equal to zero.1. For \( x^2 - 4 = 0 \) : \[ x^2 = 4 \] \[ x = \pm 2 \]2. For \( x^2 + 4 = 0 \): \[ x^2 = -4 \] \[ x = \pm \sqrt{-4} \] \[ x = \pm 2i \]This gives us the complex solutions.

Step 4: Compile the Solutions

From the above steps, we gather the real and complex solutions: 1. The real solutions are \( x = 2 \) and \( x = -2 \).2. The complex solutions are \( x = 2i \) and \( x = -2i \).

Key Concepts

Real Numbers

Real numbers are a fundamental concept in mathematics that encompass a wide range of values. They include all the values on the number line, both positive and negative, as well as zero. Here's a simple breakdown of real numbers:

• Natural numbers: These are the numbers we typically start counting from, such as 1, 2, 3, etc.
• Whole numbers: Similar to natural numbers, but they include zero: 0, 1, 2, 3, etc.
• Integers: These include whole numbers and their negative counterparts: ..., -3, -2, -1, 0, 1, 2, 3, ...
• Rational numbers: Numbers that can be expressed as fractions, like 1/2, 3, or -4.
• Irrational numbers: Numbers that cannot be expressed as a simple fraction, such as π or √2.

Real numbers include both rational and irrational numbers. They are used everywhere, from counting apples to calculating physics equations. Understanding real numbers is crucial when dealing with solutions that are real, as opposed to complex.

Difference of Squares

The difference of squares is a specific algebraic identity and is incredibly useful in algebra, particularly when factoring equations. The general form of a difference of squares is:\[ a^2 - b^2 = (a - b)(a + b) \]This identity states that if you have two squared terms separated by a subtraction sign, you can factor them into the product of the sum and difference of the square roots of those terms.
In our exercise, the equation \(x^4 - 16 = 0\) is first rewritten as \((x^2)^2 - 4^2 = 0\). Applying the difference of squares leads to \((x^2 - 4)(x^2 + 4) = 0\).
This step simplifies the problem significantly and allows it to be split into simpler expressions that are easier to solve individually. Recognizing and using the difference of squares can make solving polynomial equations much more manageable.

Factoring Equations

Factoring is the process of breaking down an equation into simpler terms (factors) that, when multiplied together, produce the original expression. It is a critical skill for solving polynomial equations. Here's an overview of how to factor equations:

• Look for common factors in all terms of the equation.
• Use identities such as the difference of squares to simplify terms.
• Rewrite the expression as a product of its factors.

In our example: \(x^4 - 16 = (x^2 - 4)(x^2 + 4) = 0\). This factoring transforms a complex fourth-degree polynomial into two simpler quadratic expressions.
Factoring is essential for finding solutions of equations since it often reduces and simplifies the problem, allowing for easier identification of the roots.

Quadratic Equations

Quadratic equations are a type of polynomial equation of the second degree. The general form is:\[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Here’s how they connect to problem-solving:

• Quadratic equations can often be solved by factoring, using the quadratic formula, or completing the square.
• They often produce two solutions which can be real or complex.

In the original equation \(x^2 - 4 = 0\), it's a straightforward quadratic that factors easily into \((x - 2)(x + 2) = 0\) with solutions \(x = 2\) and \(x = -2\). Whereas for \(x^2 + 4 = 0\), factoring doesn't work with real numbers directly, prompting the introduction of complex numbers. This approach provides solutions: \(x = 2i\) and \(x = -2i\).
Understanding quadratic equations is crucial as they appear frequently across various branches of mathematics and applied sciences.