What is the value of c so that y=x^2+9x+c is a perfect square trinomial?
A. 18
B. 9/2
C. 9/4
D. 81/4
My answer is D.
Correct answer.
( a + b ) ^ 2 = a ^ 2 + 2 a b + b ^ 2
In this case :
a = 1
2 a b = 9
2 * 1 * b = 9
2 b = 9 Divide both sides by 2
b = 9 / 2
c = b ^ 2 = ( 9 / 2 ) ^ 2 = 81 / 4
( x + 9 / 2 ) ^ 2 = x ^ 2 + 9 x + ( 81 / 4 )
Well, D looks pretty good, but let's calculate just to make sure.
To make the expression a perfect square trinomial, the coefficient of the linear term (9x) needs to be half of the coefficient of the quadratic term (x^2). So, in this case, that means c = (9/2)^2 = 81/4.
Congratulations! You were right, D is indeed the correct answer. Give yourself a pat on the back, and don't let anyone tell you that quadratic equations aren't a piece of cake. They just need a little clowning around sometimes!
To determine the value of c that makes the trinomial y = x^2 + 9x + c a perfect square trinomial, we need to compare it to the standard form of a perfect square trinomial, which is y = (x + a)^2, where a is a constant.
In this case, we have y = x^2 + 9x + c.
To find the value of c, we need to complete the square by adding and subtracting a term that will convert the given trinomial into a perfect square trinomial.
To do this, we take half of the coefficient of x, which is (9/2), square it ([(9/2)^2 = 81/4]), and add it to the equation:
y = x^2 + 9x + 81/4 + c - 81/4.
Simplifying this, we have:
y = (x + 9/2)^2 + c - 81/4.
Now, since we want the trinomial to be a perfect square trinomial, we need the term (c - 81/4) to be equal to zero. This means:
c - 81/4 = 0.
Adding 81/4 to both sides gives:
c = 81/4.
Therefore, the value of c that makes y = x^2 + 9x + c a perfect square trinomial is D. 81/4.
To determine the value of c that would make the trinomial a perfect square, we can follow these steps:
Step 1: Rewrite the trinomial in the form of a perfect square.
The general form of a perfect square trinomial is: y = (x + k)^2, where k is a constant.
So, let's start by completing the square for the trinomial y = x^2 + 9x + c.
Step 2: Take half of the coefficient of the x-term and square it.
Half of 9 is 4.5, and 4.5 squared is 20.25.
Step 3: Add the square to both sides of the equation.
y + 20.25 = x^2 + 9x + 20.25 + c
Step 4: Rewrite the right side as a perfect square trinomial.
The right side can be rewritten as (x + 4.5)^2 + c.
Step 5: The value of c that makes the trinomial a perfect square is the constant term on the right side, which is c = 0 + c. In this case, the value of c is 20.25.
Therefore, the correct option is not D (81/4), but it is actually A (18).
It is important to be careful when solving these types of questions and to carefully follow the steps to ensure accurate results.