Problem 43 - Algebra and Trigonometry

Question

Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $$ x^{2}-\frac{2}{3} x $$

Step-by-Step Solution

Answer

The constant that needs to be added to the binomial to make it a perfect square trinomial is \(\frac{1}{9}\). The resulting perfect square trinomial is \(x^2 - \frac{2}{3}x + \frac{1}{9}\), which factors into \((x - \frac{1}{3})^2\).

Step 1: Identify a and b

For the formula \(a^2 - 2ab + b^2\), we know \(a = x\) (since \(a^2 = x^2\)). We also know \(-2ab = -\frac{2}{3}x\). Solving for b, we find \(b = \frac{1}{3}\). Therefore, we can write our formula as \(x^2 - 2x*\frac{1}{3} + (\frac{1}{3})^2\).

Step 2: Determine the missing sign

Now, calculate \((\frac{1}{3})^2\), which equals \(\frac{1}{9}\). The equation becomes \(x^2 - 2x*\frac{1}{3} + \frac{1}{9}\). Hence, \(\frac{1}{9}\) is the constant to be added to the binomial to make it a perfect square trinomial.

Step 3: Factor the trinomial

Final step in this problem is to factor the trinomial. Since it is a perfect square trinomial, it factors into \((a - b)^2\). In our case, a = x and b = \frac{1}{3}, which gives us \((x - \frac{1}{3})^2\). This is the factorization of the given trinomial.

Key Concepts

Factoring Trinomials

Factoring trinomials is a fundamental skill in algebra that involves turning an expression into the product of binomials. When we have a trinomial of the form \( ax^2 + bx + c \), we often want to express it as \((mx + n)(px + q)\). This is incredibly useful for simplifying equations and finding roots.

When factoring, the goal is to find two numbers that not only multiply to give the coefficient of \( c \) but also add up to the coefficient of \( b \).

• Perfect square trinomials: These are specialized trinomials that can be expressed as the square of a binomial. They appear in the form \( a^2 - 2ab + b^2 \) and factor into \((a - b)^2\).
• Example: In the trinomial \( x^2 - \frac{2}{3}x + \frac{1}{9} \), it factors into \((x - \frac{1}{3})^2\), showing its perfect square nature.

Understanding the structure of trinomials and their factorization helps in solving more complex algebraic problems quickly and efficiently.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This technique is particularly useful in solving quadratic equations and integrating into calculus. The idea is to add a term to a binomial to make it a perfect square trinomial.

By completing the square:

• We can re-write the expression \( x^2 + bx \) as \( (x + \frac{b}{2})^2 \).
• To complete the square, you find \( \left( \frac{b}{2} \right)^2 \) and add this value to the expression.
• The original problem \( x^2 - \frac{2}{3}x \) was transformed by adding \( \left( \frac{1}{3} \right)^2 = \frac{1}{9} \), giving the perfect square.

By completing the square, students can also derive the quadratic formula or use it to convert the equation into vertex form, thereby making graphing easier.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They are the building blocks of algebra and can represent real-world quantities or purely abstract concepts. Understanding how to manipulate them is key to mastering algebra.

Some aspects of algebraic expressions include:

• Variables: Symbols like \( x \), \( y \), etc., that stand for numbers and can change.
• Constants: Numbers that do not change, like 3, -2, or \( \frac{1}{9} \).
• Operations: Addition, subtraction, multiplication, and division, which combine the variables and constants.

In working with algebraic expressions, like transforming \( x^2 - \frac{2}{3}x \) to a perfect square trinomial, students learn to appreciate the elegance and utility of algebra. They gain the ability to solve equations, model real-life situations, and perform higher-level math.