Problem 3 - Algebra and Trigonometry

Question

To complete the square of \(x^{2}-5 x\), you add the number \(\underline{\quad}.\)

Step-by-Step Solution

Answer

To complete the square of \(x^{2}-5 x\), you add the number \(6.25\).

Step 1: Identify the Coefficient of \(x\)

In the expression \(x^{2}-5 x\), the coefficient of \(x\) is -5.

Step 2: Calculate Half The Coefficient of \(x\)

Half the coefficient of \(x\) is given by \(\frac{-5}{2} = -2.5\).

Step 3: Square the Value Calculated

To complete the square, you need to add the square of the value obtained in step 2. Therefore, you square -2.5, which gives \((-2.5)^2 = 6.25\).

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental part of algebra. These equations typically have the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The variable \( x \) represents an unknown. A key characteristic of quadratic equations is the highest power of the variable, which is always 2.

Quadratic equations can depict a wide range of scenarios. From the path of a thrown ball to financial calculations, these equations are versatile. One popular method to solve quadratic equations is by "completing the square." This technique transforms the equation into a perfect square trinomial, making it easier to solve.

In this process, terms are rearranged and specific values are added to both sides. This creates a perfect square expression, simplifying the process of finding solutions. Understanding this concept is vital for diving deeper into algebra and calculus.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. For example, the expression \(x^2 - 5x\) consists of:

• The variable \(x\)
• A quadratic term \(x^2\)
• A linear term \(-5x\)

These expressions can convey complex relationships between different quantities. Manipulating them involves operations like addition, subtraction, multiplication, and division. Algebraic expressions are crucial for forming equations, which can be solved to find unknown values.

Completing the square changes an algebraic expression by transforming it into a form that's easier to work with. It's an important step in many math problems involving quadratic equations. By completing the square, you rewrite expressions in a way that reveals their properties more clearly.

Coefficient Identification

Identifying coefficients is an essential skill for working with algebraic expressions and solving equations. In a term like \(-5x\), the coefficient is the number multiplied by the variable \(x\), which is \(-5\) in this case.

When completing the square, identifying the correct coefficient is critical. This is because calculations are based on this number. Once identified, you take half of this coefficient to find a value that will complete the square.

The steps are simple but exact:

• Locate the coefficient. For \(x^2 - 5x\), the coefficient of \(x\) is \(-5\).
• Take half of the coefficient: \(\frac{-5}{2} = -2.5\).
• Square this result: \((-2.5)^2 = 6.25\).

Understanding how to identify and use coefficients properly is foundational. It ensures accuracy in transforming quadratic expressions into perfect squares, which are easier to solve and analyze.