Problem 93 - Algebra A Combined Function

Question

In your own words, explain how to calculate the number that will complete the square on an expression such as \(x^{2}-5 x\).

Step-by-Step Solution

Answer

To complete the square on \(x^2 - 5x\), add \(\frac{25}{4}\).

Step 1: Identify the Coefficient of x

The given expression is \(x^2 - 5x\). The first step is to identify the coefficient of the \(x\) term, which is \(-5\). This is the term in front of \(x\) in the expression.

Step 2: Divide the Coefficient by 2

Next, you take the coefficient of \(x\) that was identified in the previous step, which is \(-5\), and divide it by 2. This gives us \(-\frac{5}{2}\).

Step 3: Square the Result

The final step is to square the result from the previous step. So, square \(-\frac{5}{2}\) to get \(\left(-\frac{5}{2}\right)^2 = \frac{25}{4}\). This is the number that completes the square.

Key Concepts

Quadratic Expression

A quadratic expression is a type of polynomial that includes terms up to the second degree. This means it has a term like \(x^2\), but no powers higher than that. Quadratic expressions often look like \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients, and \(x\) is the variable. In the expression \(x^2 - 5x\), you can see it includes \(x^2\), which makes it a quadratic expression.

• The first term \(x^2\) represents the squared term.
• The second term \(-5x\) is the linear term.
• "Completing the square" is a technique often used to rewrite these quadratic expressions into a perfect square trinomial.

Completing the square transforms them into a form \((x + p)^2 = q\), simplifying solving for \(x\). This process is especially handy in algebra to find the roots of the quadratic equation or to make solving the equation easier.

Algebra Techniques

Algebra involves various techniques to rearrange and simplify expressions, and completing the square is just one of these methods. This technique is a structured way to change a quadratic expression into a format that is more straightforward to solve. Here’s a basic way of how this technique works:

• Identify the coefficient of the \(x\) term in the expression.
• Divide this coefficient by 2 to find half of it.
• Square the resulting number to find the constant that will complete the square.

In algebra, completing the square can be used for:

• Solving quadratic equations by making them easier to factor.
• Deriving the quadratic formula.
• Graphing quadratic functions by finding the vertex form.

These techniques require precise calculations but yield exact answers, helping students understand and manage equations better.

Coefficient in Algebra

Coefficients in algebra are the numerical or constant factors that are placed before variables in expressions or equations. They tell you how many times the variable is included in the process. In the expression \(x^2 - 5x\), you have:

• "1" as the implicit coefficient of \(x^2\) indicating one \(x^2\) term.
• "-5" as the coefficient of \(x\), explaining that \(x\) appears "+5" times but in the opposite direction.

Understanding coefficients is crucial:

• They allow manipulation of expressions to complete the square or factorize.
• Their adjustment impacts the value and behavior of the equation.
• Mastering their use helps in both simple operations and in forming complex equations.

Recognizing how coefficients work allows you to unravel the complexities of algebraic equations efficiently.