Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. x² - 10x

Verified step by step guidance

1. Identify the coefficient of the linear term (the term with x). In this case, the coefficient of x is -10.
2. Divide the coefficient of x by 2. This gives \(-10 \div 2 = -5\).
3. Square the result from the previous step. \((-5)^2 = 25\). This is the constant that should be added to the binomial to make it a perfect square trinomial.
4. Add the constant (25) to the binomial \(x^2 - 10x\), resulting in \(x^2 - 10x + 25\).
5. Factor the trinomial \(x^2 - 10x + 25\) as \((x - 5)^2\), which is the square of a binomial.

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. Recognizing this structure is essential for transforming a given binomial into a perfect square trinomial by adding the appropriate constant.

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. This technique is crucial for determining the constant needed to complete the square in the given binomial.

Factoring Quadratics

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