Problem 11 - Algebra 2

Question

Solve each equation. State the number and type of roots. \(3 x+8=0\)

Step-by-Step Solution

Answer

One real and rational root: \(x = \frac{-8}{3}\).

Step 1: Isolate the variable

The given equation is \(3x + 8 = 0\). To find the value of \(x\), we need to eliminate the constant term (\(+8\)) on the left side. This can be achieved by subtracting 8 from both sides of the equation. This gives us \(3x = -8\).

Step 2: Solve for x

Now, we need to isolate \(x\) to find its value. Since \(3x = -8\), we divide both sides of the equation by 3 to solve for \(x\). This gives us \(x = \frac{-8}{3}\).

Step 3: State the number and type of roots

Since \(x = \frac{-8}{3}\), the equation has one root. It is a rational number as it can be expressed as a fraction. Therefore, the equation has one real and rational root.

Key Concepts

Solving Equations

The process of solving equations involves finding the values of variables that satisfy the given equation. Let's break down the steps:

• First, identify and move all constant terms to the other side of the equation. We do this by applying the inverse operation - for addition, we use subtraction, and vice versa. This helps in isolating the variable.
• Next, once your variable terms are isolated, solve for the variable by performing inverse operations. If the variable is multiplied by a number, divide both sides by that number to get the variable by itself.

Remember to perform each operation on both sides of the equation to maintain balance. This process applies not only to linear equations but to equations in general.

Real Numbers

Real numbers are an essential concept in mathematics encompassing all numbers that can be found on the number line. They include:

• Natural numbers: like 1, 2, 3...
• Whole numbers: similar to natural numbers but including zero.
• Integers: which cover positive and negative numbers, including zero.
• Rational numbers: numbers like \(\frac{1}{2}\) or \(\frac{-8}{3}\) that can be expressed as fractions.
• Irrational numbers: numbers that cannot be precisely expressed as fractions, such as \(\pi \) and \(\sqrt{2}\).

In solving equations, real numbers describe the set of possible solutions or roots. The solution \(x = \frac{-8}{3}\) falls into the category of rational numbers, a subset of real numbers. Understanding this concept helps identify the type of solutions you may encounter.

Rational Root Theorem

The Rational Root Theorem is a useful tool in algebra for determining possible rational roots of a polynomial equation. This theorem implies:

• If a polynomial equation with integer coefficients has a rational solution or root, then the rational root can be expressed as \(\frac{p}{q}\), where:
• \(p\) is a factor of the constant term.
• \(q\) is a factor of the leading coefficient.

In the context of our linear equation \(3x + 8 = 0\), although solving for \(x\) doesn't directly apply the Rational Root Theorem, we can understand that the solution \(x = \frac{-8}{3}\) aligns with what we'd expect from a rational solution. Knowing this theorem can help when dealing with more complex polynomial equations.