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 + 12x

Verified step by step guidance

1. Identify the coefficient of the linear term, which is 12 in the expression \(x^2 + 12x\).
2. Divide the coefficient of the linear term by 2: \(\frac{12}{2} = 6\).
3. Square the result from the previous step: \(6^2 = 36\).
4. Add this square to the original binomial to form a perfect square trinomial: \(x^2 + 12x + 36\).
5. Factor the perfect square trinomial: \((x + 6)^2\).

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, where 'a' and 'b' are real numbers. Recognizing this structure is essential for transforming a given binomial into a perfect square trinomial.

Completing the Square

Completing the square is a method used to convert 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. This technique is crucial for solving quadratic equations and simplifying expressions.

Factoring Quadratics

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