Problem 39 - Algebra and Trigonometry

Question

Determine the constant that should be added to the binomial so that it becomes a perfect square trinomial. Then write and factor the trinomial. $$ x^{2}+3 x $$

Step-by-Step Solution

Answer

The constant needed to make the binomial a perfect square trinomial is 2.25. The trinomial is \(x^{2}+3 x+2.25\), which factors to \((x+1.5)^{2}\).

Step 1: Identify the coefficient of the x term

In the given binomial \(x^{2}+3 x\), the coefficient of the x term is 3.

Step 2: Calculate the square of half of the coefficient of x term

Dividing the coefficient of x, which is 3, by 2 gives 1.5. Then, the square of 1.5 is calculated to be 2.25.

Step 3: Add the obtained constant to the binomial to form a perfect square trinomial

Adding 2.25 to the given binomial \(x^{2}+3 x\) forms the perfect square trinomial \(x^{2}+3 x+2.25\). This is now a perfect square trinomial.

Step 4: Factorize the formed perfect square trinomial

Using the property that the perfect square trinomial \(a^{2}+2ab+b^{2}\) can be factorized into \((a+b)^{2}\), the trinomial \(x^{2}+3 x+2.25\) can be factorized into \((x+1.5)^{2}\).