Problem 281 Solve. $$ (2 x-1)^{2}=0 $$... [FREE SOLUTION]

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Chapter 6: Problem 281

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

Short Answer

Expert verified

x = \frac{1}{2}

Step by step solution

Understand the Given Equation

The equation given is $ (2x-1)^2 = 0 $. This is a quadratic equation in the form of a perfect square.

Take the Square Root of Both Sides

To eliminate the square, take the square root of both sides of the equation: \[ \sqrt{(2x-1)^2} = \sqrt{0} \]. This simplifies to \ 2x-1 = 0 \ because the square root of 0 is 0.

Solve for x

Now solve the linear equation $ 2x - 1 = 0 $. Add 1 to both sides to isolate the term with x: \[ 2x - 1 + 1 = 0 + 1 \] which simplifies to \ 2x = 1 \. Next, divide both sides by 2: \[ x = \frac{1}{2} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

perfect square

A perfect square is a number or an algebraic expression that is the square of an integer or another expression. In our problem, we see $ (2x - 1)^2 = 0 $. This indicates that \ (2x - 1) \ is a perfect square.
A perfect square has the property that it can be expressed as the product of a number or expression by itself. For example, $ 4 = 2^2 $ and $ x^2 = (x)(x) $. In our problem, $ 2x - 1 $ squared equals zero, meaning that the value inside the square must be zero.
Recognizing a perfect square helps because it indicates we can use square roots to simplify the equation.

taking square roots

Taking the square root is a method to simplify equations involving perfect squares. When both sides of the equation are squared, you can take the square root of both sides.
In the example, $ \sqrt{(2x-1)^2} = \sqrt{0} $. The square root of \ (2x-1)^2 \ is \ (2x-1) \ because the square root and the square cancel each other out. The square root of zero is zero.
So, we are left with \ 2x - 1 = 0 \. This step is crucial because it simplifies our equation and allows us to solve for x more easily.

solving linear equations

A linear equation is one of the simplest types of equations and can be solved in straightforward steps. In our problem, the equation $ 2x - 1 = 0 $ is linear.
To solve it, you need to isolate the variable, x. First, add 1 to both sides: $ 2x - 1 + 1 = 0 + 1 $, resulting in \ 2x = 1 \. Next, divide both sides by 2 to isolate x: $ \frac{2x}{2} = \frac{1}{2} $, so we get \ x = \frac{1}{2} \.
Solving linear equations requires understanding basic operations like addition, subtraction, multiplication, and division. Each step gets you closer to isolating the variable and finding its value.

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91影视

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

Short Answer

Step by step solution

Understand the Given Equation

Take the Square Root of Both Sides

Solve for x

Key Concepts

perfect square

taking square roots

solving linear equations

One App. One Place for Learning.

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