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 + 3x

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, where 'a' and 'b' are real numbers. Recognizing this structure is essential for transforming a given binomial into a perfect square trinomial by identifying the necessary constant to add.

Completing the Square

Completing the square is a method used to convert a quadratic expression into a perfect square trinomial. This involves taking half of the coefficient of the linear term, squaring it, and adding it to the expression. For the binomial x² + 3x, half of 3 is 1.5, and squaring it gives 2.25, which is the constant needed to complete the square.

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic expression as a product of its linear factors. Once the expression is transformed into a perfect square trinomial, it can be factored into the form (x + b)². This step is crucial for simplifying expressions and solving equations, as it allows for easier manipulation and understanding of the roots of the quadratic.