Problem 65 - Elementary Algebra

Question

Complete the square to make a perfect square trinomial. Then, write the result as a binomial squared. \(x^{2}-9 x\)

Step-by-Step Solution

Answer

The expression can be written as \[\left( x - \frac{9}{2} \right)^2 \]

Step 1 - Identify the quadratic and linear coefficients

Identify the coefficients of the given quadratic expression. Here, the quadratic expression is \( x^2 - 9x \). The coefficient of \( x^2 \) is 1, and the coefficient of \( x \) is -9.

Step 2 - Find the value to complete the square

Take the coefficient of \( x \), which is -9, divide it by 2, and then square the result. \[\left( \frac{-9}{2} \right)^2 = \frac{81}{4} \]

Step 3 - Add and subtract the squared value

Add and subtract the squared value inside the expression to complete the square: \[x^2 - 9x + \frac{81}{4} - \frac{81}{4} \]

Step 4 - Form the perfect square trinomial

Combine the terms to form the perfect square trinomial: \[\left( x^2 - 9x + \frac{81}{4} \right) - \frac{81}{4} \]

Step 5 - Factor the trinomial

Rewrite the trinomial as a binomial squared: \[\left( x - \frac{9}{2} \right)^2 - \frac{81}{4} \]

Step 6 - Simplify the result

Since the constant \( -\frac{81}{4} \) is not necessary for the binomial squared, focus on \[\left( x - \frac{9}{2} \right)^2 \]. Hence, the expression \[x^2 - 9x \] can be written as \[\left( x - \frac{9}{2} \right)^2 \]

Key Concepts

Quadratic Expression

A quadratic expression is a polynomial expression of degree 2, which typically follows the form \(ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants, with \(a\) not equal to zero. In our exercise, the quadratic expression given is \(x^2 - 9x\), where the coefficients are 1 (for \(x^2\)) and -9 (for \(x\)). Recognizing these coefficients helps us in the process of completing the square. This process rearranges the expression into a perfect square trinomial, making it easier to solve quadratics later.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. These trinomials have the form \((x + a)^2\) or \((x - a)^2\). For example, if we have \(x^2 - 9x\), we start the process by identifying that our goal is to convert it into a perfect square trinomial. Following the steps in the provided solution:

• We take the linear coefficient (which is -9), divide it by 2, and square the result: \(\left(\frac{-9}{2}\right)^2 = \frac{81}{4}\).
• Next, we add and subtract this squared value within the expression, rewriting \(x^2 - 9x\) as \(x^2 - 9x + \frac{81}{4} - \frac{81}{4}\). This step is crucial because it helps to balance the equation, keeping it equivalent to the original expression.
• By grouping \(x^2 - 9x + \frac{81}{4}\) together, we have formed a perfect square trinomial.

This approach transforms the initial expression into something more manageable and sets us up for the next step: factoring.

Factoring

Factoring is the process of breaking down an expression into a product of simpler expressions, or factors. In the case of completing the square, once we've rewritten our quadratic expression as a perfect square trinomial, we can factor it as a binomial squared. From our work:

• We identified the perfect square trinomial: \(x^2 - 9x + \frac{81}{4}\).
• This can be rewritten as \((x - \frac{9}{2})^2\).

When factoring in this manner, it simplifies solving equations because the expression is now represented in an easy-to-solve binomial squared form. The final expression, \((x - \frac{9}{2})^2\), indicates that our original quadratic expression, \(x^2 - 9x\), can be factored and thus solved more efficiently. Factoring using the completing-the-square method is particularly useful for quadratic equations that do not factor neatly by simpler methods.