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. $x^2-\frac{2}{3}x$

Verified step by step guidance

1. Identify the coefficient of the linear term in the binomial. Here, the binomial is \(x^2 - \frac{2}{3}x\), so the coefficient of \(x\) is \(-\frac{2}{3}\).
2. Take half of the coefficient of \(x\). Half of \(-\frac{2}{3}\) is \(-\frac{1}{3}\).
3. Square the result from step 2. Squaring \(-\frac{1}{3}\) gives \(\left(-\frac{1}{3}\right)^2 = \frac{1}{9}\).
4. Add this squared value to the original binomial to form a perfect square trinomial: \(x^2 - \frac{2}{3}x + \frac{1}{9}\).
5. Write the trinomial as a squared binomial using the form \((x + a)^2 = x^2 + 2ax + a^2\). Here, it factors as \(\left(x - \frac{1}{3}\right)^2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial, typically in the form (a + b)^2 = a^2 + 2ab + b^2. Recognizing or creating such trinomials helps simplify factoring and solving quadratic expressions.

Completing the Square

Completing the square involves adding a constant term to a quadratic expression to form a perfect square trinomial. This constant is found by taking half the coefficient of the linear term, squaring it, and adding it to the expression, enabling easier factoring or solving.

Factoring Quadratic Expressions

Factoring quadratic expressions means rewriting them as a product of binomials or squares of binomials. After completing the square, the trinomial can be factored into (x + d)^2 form, which simplifies solving equations or analyzing the function.