Problem 78 - Algebra 1: Concepts and Skills

Question

What term should you add to \(x^{2}-\frac{1}{2} x\) to create a perfect square trinomial? F. \(\frac{1}{2}\) G. \(\frac{1}{4}\) H. \(\frac{1}{16}\) J. \(\frac{1}{32}\)

Step-by-Step Solution

Answer

The term that should be added to the expression to create a perfect square trinomial is \(\frac{1}{16}\) , thus the right answer is H. \(\frac{1}{16}\)

Step 1: Identify the Coefficient

The coefficient of the \(x\) term in the expression \(x^{2}-\frac{1}{2} x\) is \(-\frac{1}{2}\)

Step 2: Divide by Two and Square the Result

Take the coefficient of the \(x\) term, divide it by 2 and then square the result. Thus, \(\left(-\frac{1}{2}\right)/2 = -\frac{1}{4}\) and \(\left(-\frac{1}{4}\right)^{2}=\frac{1}{16}\)

Step 3: Result

Therefore, the term that should be added to \(x^{2}-\frac{1}{2} x\) to create a perfect square trinomial is \(\frac{1}{16}\)

Key Concepts

Completing the Square

Completing the square is a powerful technique in algebra that allows us to convert a quadratic expression into a perfect square trinomial. This method involves a series of algebraic manipulations to reshape the quadratic expression into a form that can be easily factored or used to solve quadratic equations. The basic steps include: (1) ensuring the coefficient of the quadratic term is one - if not, we factor it out, (2) taking half the coefficient of the linear term, squaring it, and adding this square to both sides of the equation or to the expression, and (3) writing the expression as the square of a binomial.

For example, consider the expression \(x^2 - \frac{1}{2}x\). To complete the square for this expression, we take the coefficient of \(x\), which is \(\frac{1}{2}\), divide it by two to get \(\frac{1}{4}\), and then square the result to find \(\frac{1}{16}\), which we add to the original expression to form the perfect square trinomial \(x^2 - \frac{1}{2}x + \frac{1}{16}\). This is advantageous for solving quadratic equations and understanding the properties of parabolas.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations like addition, subtraction, multiplication, and division. For instance, \(x^2 - \frac{1}{2}x\) is an algebraic expression that includes a quadratic term \(x^2\) and a linear term \(\frac{1}{2}x\).

Understanding how to manipulate these expressions is crucial in algebra. This involves skills such as combining like terms, expanding products, and factoring. When we work with expressions like the one given, it's important to recognize the terms and coefficients in order to apply methods such as completing the square effectively. Being able to rewrite expressions into different forms can offer insights into their properties and help to solve algebraic equations.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally written in the form \(ax^2 + bx + c = 0\). Solving these equations is a fundamental aspect of algebra. One way to solve a quadratic equation is by completing the square, which transforms the equation into a perfect square trinomial, revealing the roots of the equation.

The equation we derive by completing the square can then be solved by taking the square root of both sides, leading to two possible solutions for \(x\). In the context of the given expression \(x^2 - \frac{1}{2}x\), if we set up an equation \(x^2 - \frac{1}{2}x + \frac{1}{16} = 0\), completing the square would enable us to solve for \(x\) by finding values that satisfy the equation.

Polynomial Arithmetic

Polynomial arithmetic refers to the operations performed on polynomials, including addition, subtraction, multiplication, and division. The arithmetic is governed by the distributive, associative, and commutative properties. Understanding polynomial arithmetic is essential for operations like completing the square, factoring, and simplifying expressions.

When we add the term \(\frac{1}{16}\) to the expression \(x^2 - \frac{1}{2}x\), we're using polynomial addition to form \(x^2 - \frac{1}{2}x + \frac{1}{16}\), which now represents a perfect square trinomial. Grasping the basics of polynomial arithmetic unlocks the ability to work with more complex algebraic structures and to solve a variety of mathematical problems.