Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $x squared plus 20 x$

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Identify the binomial given: $x^2 + 20x$.

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 $x$.

Calculate $\left(\frac{20}{2}\right)^2 = 10^2 = 100$. This is the constant that should be added to the binomial.

Write the perfect square trinomial by adding 100: $x^2 + 20x + 100$.

Factor the trinomial as a square of a binomial: $\left(x + 10\right)^2$.

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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 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.

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+20x x^2+ 20x