Problem 30 - College Algebra Essentials

Question

Solve each equation in Exercises \(15-34\) by the square root property. $$ (4 x-1)^{2}=16 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 5/4, -3/4\)

Step 1: Rearrange the equation

First, isolate the square term on one side by making the equation to look like this: \( (4x-1)^2 = 16 \)

Step 2: Applying the Square Root Property

Apply the square root property to both sides of the equation. This implies that \( 4x-1 = \pm \sqrt{16} \). Remember to include both the positive and negative square roots when performing this step.

Step 3: Simplify the square root

Simplify the square root of 16 which is \(\pm 4\). This gives \( 4x-1 = \pm 4 \).

Step 4: Solving for x

We then solve for the variable \(x\). This gives two potential solutions one for \(4x - 1 = 4\) as \(x = 5/4\) and one for \(4x - 1 = -4\) as \(x = -3/4\).

Key Concepts

Equation Solving

Equation solving is a process that allows us to find the unknown values in an equation that make the equation true. Whether you're working with a basic arithmetic equation or more complex algebraic equations, the goal is always the same: to isolate the variable.

• Identify the variable: This is usually what you're trying to find. In our example, it's \(x\).
• Use inverse operations: These are operations that reverse the effect of another operation. For instance, if a term on one side of the equation is being added, you'll subtract that same term to undo the addition.
• Maintain balance: An equation is like a balanced scale. Whatever operation you do on one side must be done to the other to keep it balanced. This is crucial for arriving at the correct solution.

By systematically applying these steps, you'll be able to solve not just quadratic equations using the square root property, but also many other types of equations.

Algebraic Equations

Algebraic equations are equations formed by combining numbers and algebraic expressions. They represent relationships between different quantities and are often used to model real-world situations. An algebraic equation can contain constants, variables, coefficients, and operators such as addition, subtraction, multiplication, and division.
A key aspect of working with algebraic equations is understanding the order of operations and the properties of equality, which help in manipulating and simplifying the equations effectively.

• Constants: Fixed values that don't change. In our problem, '16' is a constant.
• Variables: Symbols that represent unknown values, such as 'x' in our equation.
• Coefficients: Numbers multiplied by the variables, such as '4' in \(4x\).
• Operators: Symbols that represent mathematical operations, like '+', '-', '*', or '/'.

Understanding how these components interact is essential for solving algebraic equations accurately.

Quadratic Equations

Quadratic equations are a special type of algebraic equation where the highest degree of the variable is two. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.
Our equation \((4x-1)^2 = 16\) can be seen as a disguised form of a quadratic equation. Here, solving by the square root property involves taking the square root of both sides to simplify the quadratic expression.
The square root property allows us to directly find the solutions by recognizing that if \((n)^2 = k\), then \(n = \pm \sqrt{k}\). This can often lead us to find two solutions:

• A positive solution
• A negative solution

When working with quadratics, always remember to consider both roots, as they will give you different perspectives on the solutions present in the equation.