Problem 31 - Precalculus with Limits

Question

In Exercises 23-32, find the zeros of the function algebraically. \(f(x) = \sqrt{2x} - 1\)

Step-by-Step Solution

Answer

The zeros of the function are x = 0.5.

Step 1: Set up the equation

First, the function \(f(x) = \sqrt{2x} - 1\) is set to zero: \[0 = \sqrt{2x} - 1\].

Step 2: Isolate the square root term

Next, isolate the square root term on one side of the equation by adding 1 to both sides: \[1 = \sqrt{2x}\].

Step 3: Square both sides

Then, square both sides of the equation to remove the square root: \[1^2 = (\sqrt{2x})^2 \rightarrow 1 = 2x\].

Step 4: Solve for x

Finally, divide each side by 2 to solve for x: \[x = 0.5\]

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations to make them easier to solve. It's a foundational skill in mathematics that allows us to isolate variables and clear complex terms, facilitating an understanding of more intricate problems later on. In this exercise, we began by setting the function equal to zero:

• This action is based on the principle that finding zeros of a function, meaning where the function equals zero, reveals the roots or solutions of the function.

Next, we performed the algebraic manipulation of isolating the square root term by adding 1 to both sides.
This is an important step because it simplifies the equation into a form where the variable is only under the square root, making it easier to handle in future steps.
Understanding that each move in algebra aims to simplify equations step-by-step is crucial.

• Algebraic manipulation is not only about knowing what steps to take but also understanding why they are essential in reaching a solution.

Solving Equations

Being proficient in solving equations is a must for tackling algebra problems. Solving equations involves finding all possible values of the variable that make the equation true. In our example, after isolating the square root term

• The next logical step was to eliminate the square root by squaring both sides of the equation.

This gives us a new equation that is inherently easier to solve. Squaring both sides was necessary because it converted the square root equation into a linear equation.

• The resulting equation is straightforward: \[1 = 2x\]
• With linear equations, the goal is simply to get the variable by itself on one side.

This makes it clear how each previous manipulation leads into simple arithmetic operations to finally solve for the variable.

Square Root Equations

Understanding square root equations is vital as they frequently pop up in algebra and calculus. A square root equation is an equation in which the variable is inside a square root.
These kinds of equations can initially seem complex, but breaking them down into manageable steps makes them solvable. To simplify and solve a square root equation:

• First, isolate the square root on one side, as done in the step: \[1 = \sqrt{2x}\]
• Then, eliminate the square root by squaring both sides, which transforms the equation to linear form.

Remember to always check any solutions back in the original equation.
This ensures that the original condition is met since squaring both sides can sometimes introduce extraneous solutions.
Thus, solving square root equations not only involves mathematical operations but also verifying solutions for consistency.