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. $x^2-7x$

Verified step by step guidance

1. Identify the coefficient of the linear term in the binomial. Here, the binomial is \(x^2 - 7x\), so the coefficient of \(x\) is \(-7\).
2. Take half of the coefficient of \(x\). Calculate \(\frac{-7}{2}\), which is \(-\frac{7}{2}\).
3. Square the result from step 2. Compute \(\left(-\frac{7}{2}\right)^2 = \frac{49}{4}\).
4. Add this squared value to the original binomial to form a perfect square trinomial: \(x^2 - 7x + \frac{49}{4}\).
5. Write the trinomial as a squared binomial using the form \(\left(x + a\right)^2 = x^2 + 2ax + a^2\). Here, it factors as \(\left(x - \frac{7}{2}\right)^2\).

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 to a quadratic expression to form a perfect square trinomial. This constant is found by taking half the coefficient of the x-term, 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 a binomial squared. Once the perfect square trinomial is formed, it can be factored into (x + a)^2 or (x - a)^2, simplifying solving or further manipulation.