In Exercises 64–65, 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+ 20x

Verified step by step guidance

1. Identify the coefficient of the linear term in the binomial, which is 20 in the expression x^2 + 20x.
2. Divide the coefficient of the linear term by 2, then square the result to find the constant that should be added. This is calculated as (20/2)^2.
3. Add the calculated constant to the original binomial to form a perfect square trinomial. The new expression will be x^2 + 20x + (20/2)^2.
4. Factor the perfect square trinomial. It can be factored as (x + the square root of the constant)^2.
5. Write the factored form of the trinomial using the value calculated in step 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 quadratic expression 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 the coefficient of the linear term. For the expression x² + 20x, the constant to be added is (20/2)² = 100, allowing us to rewrite the expression as (x + 10)².

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic expression as a product of its linear factors. For a perfect square trinomial like (x + 10)², it can be factored as (x + 10)(x + 10). Understanding how to factor is crucial for simplifying expressions and solving quadratic equations effectively.