Question

\(\frac{x}{2}=\frac{2\left(n^{0}\right)}{2}-\frac{1}{2 x}\) What is the value of \(x\) ? (A) 0 (B) 1 (C) 2 (D) 3

Short answer

The value of \( x \) is 1.

Step-by-step solution

Simplify the Right Side of Equation

The right side of the equation is \( \frac{2(n^0)}{2} - \frac{1}{2x} \). Since \( n^0 = 1 \), it becomes \( \frac{2 \cdot 1}{2} - \frac{1}{2x} \), which simplifies to \( 1 - \frac{1}{2x} \).

Set Both Sides Equal

The equation \( \frac{x}{2} = 1 - \frac{1}{2x} \) has both sides simplified, and we need to solve for \( x \).

Clear Fractions by Multiplying Through by 2x

Multiply both sides by \( 2x \) to eliminate the fractions: \( x(x) = 2x(1) - 1 \). This simplification results in \( x^2 = 2x - 1 \).

Rearrange the Equation

Rearrange the equation to standard quadratic form: \( x^2 - 2x + 1 = 0 \).

Factor the Quadratic Equation

The equation \( x^2 - 2x + 1 = 0 \) can be factored as \( (x - 1)(x - 1) = 0 \) or \( (x - 1)^2 = 0 \).

Solve for x

Solve \( (x - 1)^2 = 0 \) by taking the square root of both sides, giving \( x - 1 = 0 \). Thus, \( x = 1 \).

Key concepts

Solving Equations

Solving equations is a fundamental skill in mathematics that allows us to find unknown values. To solve an equation means to find all possible values of the variable that makes the equation true. Here's how we approach it:

• Identify the equation you need to solve. In our example, it starts as: \( \frac{x}{2} = 1 - \frac{1}{2x} \).
• First, simplify any complex terms or expressions. Notice that on the right-hand side, we simplified the term \( \frac{2(n^0)}{2} \) to \( 1 \).
• Isolate the variable by clearing out fractions and simplifying, leading to a more workable form. To do this, multiplying every term by the lowest common denominator helps clear the fractions, transforming our example into \( x^2 = 2x - 1 \).
• Continue simplifying until the equation is in one of the standard forms like linear or quadratic, making it easier to solve for the unknown variable.

By following these steps methodically, you can systematically solve for the unknowns in any equation.

Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. The standard form is given by \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants.

Here's how to work with quadratic equations:

• The first step in solving a quadratic equation is to ensure it is in the standard form by rearranging terms if necessary. In our exercise, we rearranged \( x^2 - 2x + 1 = 0 \).
• Once in standard form, look for factoring options, if applicable. Factoring is the process of breaking down an equation into simpler components. Our example factors to \( (x - 1)^2 = 0 \).
• Quadratics can also be solved using the quadratic formula or by completing the square if factoring is not straightforward. But when the equation easily factors like ours, it's the simplest path to solving it.

The factorized form gives us the solutions directly, making it a powerful method for solving quadratic equations.

Problem Solving

Problem solving in mathematics involves using logical reasoning and analytical skills to find solutions.

To solve a problem like the presented equation, consider these key strategies:

• Understand and simplify the problem. Break it into smaller parts. For example, simplifying each side of the equation helps.
• Translate the problem into a mathematical equation or set of equations. Our original exercise became a quadratic equation, which struck at the core of solving it.
• Choose the appropriate strategy based on the form of the equation. With quadratics, knowing methods like factoring or the quadratic formula equips you to solve efficiently.
• Verify the solution's correctness. After solving \( (x - 1)^2 = 0 \), verify by substituting back into the original equation: \( \frac{1}{2} = 1 - \frac{1}{2 \times 1} \).

These steps collectively guide you through successfully solving complex problems, making use of systematic methods and mathematical reasoning.