In Exercises 35–46, 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 - (2/3)x

Verified step by step guidance

1. Identify the coefficient of the linear term, which is $-\frac{2}{3}$.
2. Divide this coefficient by 2: $-\frac{2}{3}÷2=-\frac{1}{3}$.
3. Square the result from step 2: $(-\frac{1}{3}{)}^2=\frac{1}{9}$.
4. Add this square to the binomial to form a perfect square trinomial: $x^2-\frac{2}{3}x+\frac{1}{9}$.
5. Factor the trinomial as a square of a binomial: $(x-\frac{1}{3}{)}^2$.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be expressed as the square of a binomial. It takes the form (a ± b)² = a² ± 2ab + b². Recognizing this structure is essential for determining the constant needed to complete the square in a given expression.

Completing the Square

Completing the square is a method used to transform a quadratic expression into a perfect square trinomial. This involves adding a specific constant to the expression, which is derived from taking half of the coefficient of the linear term, squaring it, and adding it to the original expression to maintain equality.

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic expression as a product of its linear factors. For a perfect square trinomial, this means expressing it in the form (a ± b)². Understanding how to factor is crucial for simplifying expressions and solving quadratic equations.