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 + 3x

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1. To make the binomial \(x^2 + 3x\) a perfect square trinomial, we need to add a constant. Recall that a perfect square trinomial takes the form \((x + a)^2 = x^2 + 2ax + a^2\).
2. Compare \(x^2 + 3x\) with \(x^2 + 2ax\). Here, \(2a = 3\). Solve for \(a\) by dividing both sides of the equation by 2: \(a = \frac{3}{2}\).
3. The constant to add is \(a^2\). Substitute \(a = \frac{3}{2}\) into \(a^2\): \(a^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4}\).
4. Add \(\frac{9}{4}\) to the binomial \(x^2 + 3x\) to form the perfect square trinomial: \(x^2 + 3x + \frac{9}{4}\).
5. Factor the trinomial \(x^2 + 3x + \frac{9}{4}\) as \(\left(x + \frac{3}{2}\right)^2\).

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, where 'a' and 'b' are real numbers. 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. For the binomial x² + 3x, half of 3 is 1.5, and squaring it gives 2.25, which is the constant needed to complete the square.

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic expression as a product of its linear factors. Once a quadratic is expressed as a perfect square trinomial, it can be factored into the form (x + b)². For the expression x² + 3x + 2.25, it factors to (x + 1.5)², demonstrating the relationship between the coefficients and the roots of the quadratic.