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. $x squared minus 7 x$

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Identify the coefficient of the linear term in the binomial. Here, the binomial is $x^2 - 7x$, so the coefficient of $x$ is $-7$.

Take half of the coefficient of $x$. Calculate $\frac{-7}{2}$, which is $-\frac{7}{2}$.

Square the result from step 2. Compute $\left(-\frac{7}{2}\right)^2 = \frac{49}{4}$.

Add this squared value to the original binomial to form a perfect square trinomial: $x^2 - 7x + \frac{49}{4}$.

Write the trinomial as a squared binomial using the form $\left(x + a\right)^2 = x^2 + 2ax + a^2$. Here, it factors as $\left(x - \frac{7}{2}\right)^2$.

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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.

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. x2−7xx^2 - 7x