Problem 20 - Algebra 1

Question

Find the term that should be added to the expression to create a perfect square trinomial. $$x^{2}-12 x$$

Step-by-Step Solution

Answer

The term that should be added to the expression \(x^{2}-12 x\) to create a perfect square trinomial is \(36\).

Step 1: Identify the binomial

The given binomial that we want to complete is \(x^{2} - 12x\). We can compare it with the general form \(a^2 - 2ab\). Here we can see that \(a = x\) and \(2ab = 12x\).

Step 2: Find the value of 'b'

From Step 1, we know that \(2ab = 12x\). We also know \(a = x\). Substitute \(a = x\) in the equation, we'll get \(2b * x = 12x\). Dividing both sides by \(2x\) gives us \(b = \frac{12x}{2x} = 6\).

Step 3: Determine the square of 'b'

We now have \(b = 6\). The element that will make our quadratic binomial a perfect square trinomial is \(b^2\). Therefore, we find the square of \(b: b^2 = 6^2 = 36\). This is the value we need to add to the given expression to complete the square.

Key Concepts

Binomial

A binomial is an algebraic expression containing exactly two terms. It's a specific type of polynomial that you'll often encounter in algebra. In our exercise, the expression \(x^2 - 12x\) serves as a typical example of a binomial. Binomials are essential in algebra because they're often a foundation for more complex expressions, such as trinomials and other polynomials.

Here are a few key points about binomials:

• Contains two terms separated by either a plus or minus sign, e.g., \(ax + b\).
• Can include variables, coefficients, and constant terms.
• Are used frequently to form quadratic equations and trinomials.

Understanding binomials is crucial as it allows you to manipulate expressions easily, set the stage for solving equations, and derive further mathematical results.

Completing the Square

Completing the square is a technique used to convert a quadratic expression into a perfect square trinomial. This is useful in solving quadratic equations, graphing parabolas, and simplifying algebraic expressions. The process involves manipulating the expression to reveal a perfect square.

The steps to complete the square are:

• Identify the binomial you want to transform. In the exercise, it's \(x^2 - 12x\).
• Determine the term to add and make it a perfect square. First, find the missing value by halving the coefficient of \(x\) and then squaring it. For \(x^2 - 12x\), \(\frac{-12}{2} = -6\), and its square is \((-6)^2 = 36\).
• Add this square to the binomial to form the perfect square trinomial. Hence, the expression becomes \(x^2 - 12x + 36\).

This method simplifies many quadratic problems, providing a straightforward way to factor expressions or solve equations.

Quadratic Expression

A quadratic expression is a polynomial where the highest degree of any term is two. Quadratics are fundamental in algebra due to their wide use in various mathematical fields. The exercise involves transforming the binomial \(x^2 - 12x\) into a quadratic expression that represents a perfect square trinomial.

Quadratic expressions generally have the form \(ax^2 + bx + c\):

• \(a\), \(b\), and \(c\) are constants, with \(a eq 0\).
• They can be represented graphically, typically forming a parabola on a Cartesian plane.
• Quadratics can be solved through factoring, completing the square, or using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).

By converting our initial binomial into a perfect square trinomial, we effectively reshape it into a complete quadratic expression. This facilitates easier manipulation for further operations such as solving for variable values or determining graph characteristics.