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. $x^2-\left(1/3\right)x$

Verified step by step guidance

1. Identify the coefficient of the linear term in the binomial. Here, the binomial is \(x^2 - \frac{1}{3}x\), so the coefficient of \(x\) is \(-\frac{1}{3}\).
2. To complete the square, take half of the coefficient of \(x\). Calculate \(\frac{1}{2} \times \left(-\frac{1}{3}\right) = -\frac{1}{6}\).
3. Square the result from the previous step to find the constant to add: \(\left(-\frac{1}{6}\right)^2 = \frac{1}{36}\).
4. Add this constant to the original binomial to form the perfect square trinomial: \(x^2 - \frac{1}{3}x + \frac{1}{36}\).
5. Write the trinomial as a squared binomial using the value found in step 2: \(\left(x - \frac{1}{6}\right)^2\).

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 this form helps in rewriting and factoring quadratics efficiently.

Completing the Square

Completing the square involves adding a constant 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.

Factoring Quadratic Expressions

Factoring quadratic expressions means rewriting them as a product of binomials. Once a perfect square trinomial is formed, it can be factored as the square of a binomial, simplifying solving or further manipulation.