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. $x squared plus 12 x$

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Identify the coefficient of the linear term in the binomial. Here, the binomial is $x^2 + 12x$, so the coefficient of $x$ is 12.

To complete the square, take half of the coefficient of $x$, which is $\frac{12}{2} = 6$.

Square this value: $6^2 = 36$. This is the constant that should be added to the binomial to make it a perfect square trinomial.

Add the constant to the binomial to form the trinomial: $x^2 + 12x + 36$.

Write the trinomial as a squared binomial: $(x + 6)^2$.

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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 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. For perfect square trinomials, this process simplifies to expressing the trinomial as (x + a)^2, which is useful for solving equations and analyzing functions.

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+12xx^2 + 12x