Problem 323 - Elementary Algebra

Question

Solve. \((2 x-1)^{2}=0\)

Step-by-Step Solution

Answer

x = \frac{1}{2}

Step 1 - Expand the Equation

First, recognize that the equation \( (2x - 1)^2 = 0 \) is already simplified. No need to expand further.

Step 2 - Take the Square Root of Both Sides

Take the square root of both sides to eliminate the exponent: \ \sqrt{(2x - 1)^2} = \sqrt{0} \. This results in \(2x - 1 = 0 \).

Step 3 - Solve for x

Solve for \(x\) by isolating it: \ 2x - 1 = 0 \. Add 1 to both sides: \ 2x = 1 \. Then, divide both sides by 2: \ x = \frac{1}{2} \.

Key Concepts

Equation Simplification

When solving quadratic equations, the first step often involves simplifying the equation. This means to reduce any complex expressions into a more manageable form. In the case of \( (2x - 1)^2 = 0 \), the equation is already simplified.
Here are some key points to remember:

• Check if any terms can be combined or canceled.
• Ensure that all like terms (terms with the same variable and exponent) are together.
• Simplification makes it easier to solve the equation in later steps.

By simplifying, you prepare the equation for the next steps, such as applying the square root property or solving for a variable.

Square Root Property

The square root property helps in solving equations where a variable is squared. The general concept is that if you have an equation of the form \(a^2 = b \), taking the square root of both sides gets you back to a linear equation.
For example, in the equation \( (2x - 1)^2 = 0 \):

• Use the square root on both sides: \ \sqrt{(2x - 1)^2} = \sqrt{0} \.
• This simplifies to: \(2x - 1 = 0 \), as the square root and the square cancel out.
• With \( \sqrt{0} \) being 0, you now have a simpler equation to solve.

Taking the square root simplifies the problem and allows us to solve for the variable in a straightforward manner.

Linear Equation

A linear equation is an equation of the first degree, meaning it has no exponents greater than one. It can generally be written in the form \( ax + b = c \). Solving linear equations often involves isolating the variable. Here’s how you would approach this:

• First, we have \( 2x - 1 = 0 \) from the previous step.
• Add 1 to both sides: \( 2x = 1 \).
• Divide both sides by 2: \ x = \frac{1}{2} \.

The result is \( x = \frac{1}{2} \), which is the solution to the original equation. Linear equations are fundamental and make it possible to solve more complex algebraic problems.