Problem 26 - Algebra 2

Question

Find the value of c that makes each trinomial a perfect square. Then write the trinomial as a perfect square. \(x^{2}-15 x+c\)

Step-by-Step Solution

Answer

The value of c is \(\frac{225}{4}\), and the trinomial is \((x - \frac{15}{2})^2\).

Step 1: Understand the Form of a Perfect Square Trinomial

A perfect square trinomial is one that can be expressed as \((x - a)^2\) or \((x + a)^2\). For the expression \(x^2 - 15x + c\), we want it to match the form \(x^2 - 2ax + a^2\).

Step 2: Identify '2a' and 'a'

In the expression \(x^2 - 15x + c\), compare the term \(-15x\)\ with \(-2ax\)\ to find \(a\). Here, \(-2a = -15\), so \(a = \frac{15}{2}\).

Step 3: Calculate 'c'

With \(a = \frac{15}{2}\), calculate \(c\) as \(a^2\). Therefore, \(c = \left(\frac{15}{2}\right)^2 = \frac{225}{4}\).

Step 4: Write the Trinomial as a Perfect Square

Now, substitute \(c = \frac{225}{4}\) back into the trinomial to confirm it can be written as a perfect square. Hence, \(x^2 - 15x + \frac{225}{4} = \left(x - \frac{15}{2}\right)^2\).

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomial expressions where the highest exponent of the variable is 2. These expressions often take the form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Let's break down this formula:

• \(a\) is the coefficient of the quadratic term \(x^2\).
• \(b\) is the coefficient of the linear term \(x\).
• \(c\) is the constant term, which doesn't involve \(x\).

When specifically dealing with a perfect square trinomial, the expression can be represented as \((x+a)^2 = x^2 + 2ax + a^2\) or \((x-a)^2 = x^2 - 2ax + a^2\). Identifying this structure helps in simplifying quadratic expressions and solving equations. Perfect square trinomials have neat and predictable outcomes, making them a special category of quadratic expressions.

Completing the Square

Completing the square is a method used in algebra to transform a quadratic expression into a perfect square trinomial. This technique is valuable for solving quadratic equations and is a vital step in deriving the quadratic formula.To complete the square for an expression like \(x^2 - bx + c\), follow these steps:

• Take the coefficient of the linear term \(b\), divide it by 2 to find \(a\).
• Square \(a\) to find the constant \(c\) that completes the square.
• Add this \(c\) to the expression, rewriting it in the completed square form.

This method allows the quadratic expression to be expressed as \((x - a)^2\), making it easier for manipulation and solution. For the trinomial \(x^2 - 15x + c\), identifying that \(a = \frac{15}{2}\) and \(c = \left(\frac{15}{2}\right)^2\) ensures it becomes a perfect square trinomial.

Polynomial Equations

Polynomial equations involve sums of terms that consist of variables raised to whole number powers, with each term having zero or more variables. Quadratic expressions are a subset of polynomial equations, focusing specifically on equations with the highest power of 2.Understanding the structure of polynomial equations is essential:

• Each term in the polynomial is a simple combination of a variable and a coefficient.
• The degree of the equation is determined by the highest power of the variable in the expression.
• Solutions to polynomial equations correspond to the values of the variable that make the equation zero.

Converting a quadratic equation into a perfect square form is a strategic approach. It simplifies solving by reducing the complexity of the polynomial. For instance, converting \(x^2 - 15x + \frac{225}{4}\) to \((x - \frac{15}{2})^2\) illustrates this approach. It demonstrates how understanding the relationship between coefficients and terms aids in both simplifying and solving polynomial equations.