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 binomial given: \(x^2 + 20x\).
2. Recall that to complete the square, we add a constant term to make the expression a perfect square trinomial. The formula for the constant to add is \(\left(\frac{b}{2}\right)^2\), where \(b\) is the coefficient of \(x\).
3. Calculate \(\left(\frac{20}{2}\right)^2 = 10^2 = 100\). This is the constant that should be added to the binomial.
4. Write the perfect square trinomial by adding 100: \(x^2 + 20x + 100\).
5. Factor the trinomial as a square of a binomial: \(\left(x + 10\right)^2\).

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 (x + a)^2 = x^2 + 2ax + a^2. Recognizing this form helps in rewriting and factoring quadratics efficiently.

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 x, then squaring it, which allows the expression to be factored as a binomial squared.

Factoring Quadratic Expressions

Factoring quadratic expressions means rewriting them as a product of binomials or squares of binomials. Once a perfect square trinomial is formed, it can be factored into (x + a)^2, simplifying solving or further manipulation.