Complete the square. Write the resulting perfect square trinomial as the square of a binomial. X2+9x
x^2 + 9x
= x^2 + 9x + 81/4 - 81/4 , take 1/2 of the coefficient of the x term, square it, then add and subtract it
= (x + 9/2)^2 - 81/4
Ah, completing the square! It's like giving a trinomial a little makeover. Let's see what we can do with X^2 + 9x.
To complete the square, we need to add and subtract (b/2)^2, where b is the coefficient of our linear term. In this case, b = 9, so (b/2)^2 would be (9/2)^2 or 81/4.
Adding and subtracting 81/4, we get:
X^2 + 9x + 81/4 - 81/4
Now, let's group this into perfect square trinomials:
(X^2 + 9x + 81/4) - 81/4
Notice that the first three terms form a perfect square trinomial, which can be factored as (X + 9/2)^2. Simplifying the last term, we have:
(X + 9/2)^2 - 81/4
So, the resulting perfect square trinomial can be written as the square of a binomial: (X + 9/2)^2. Voila!
To complete the square, follow these steps:
Step 1: Take half of the coefficient of x and square it:
c = (9/2)^2 = 81/4
Step 2: Add the result from step 1 to both sides of the equation:
x^2 + 9x + 81/4 = x^2 + 9x + 81/4 + 81/4
Step 3: Simplify the right side of the equation:
x^2 + 9x + 81/4 = x^2 + 9x + 162/4
Step 4: Factor the left side of the equation:
(x + 9/2)^2 = x^2 + 9x + 81/4
So, the perfect square trinomial x^2 + 9x can be written as the square of the binomial (x + 9/2)^2.
To complete the square for the trinomial x^2 + 9x, you can follow these steps:
Step 1: Divide the coefficient of the x term by 2 and square the result:
(9 / 2)^2 = 81 / 4
Step 2: Add the result from Step 1 to the expression, both inside and outside the parentheses:
x^2 + 9x + 81/4 - 81/4
Step 3: Rearrange the expression:
(x^2 + 9x + 81/4) - 81/4
Step 4: Factor the perfect square trinomial inside the parentheses:
(x + 9/2)^2 - 81/4
So, the resulting perfect square trinomial is (x + 9/2)^2.