Question

What is the value of c so that x^2+8x+c is a perfect square trinomial?

Answer 1

What makes a perfect square trinomial?

Answer 2

You want to factor x^2 + 8x + c into the form (x+a)^2, where 'a' is a number that multiplies together to equal 'c', but also adds together to equal 8.

So, we can form a system of linear equations:
(1) a * a = c
(2) a + a = 8

The second of the two tells us that a = 4.

Plugging that back into the first tells us 4 * 4 = 16 = c.

So x^2 + 8x + 16 = (x + 4)^2

Answer 3

So it's a perfect square trinomial when what?

Answer 4

@TurtleMuffin

Answer 5

Ah sorry! I'm new to this site and still not familiar with how it works. Didn't see your response.

To be honest I didn't even know what a perfect square trinominal was before I read this question. I had to look it up. Vocabulary in mathematics isn't as important as your teachers may make it out to. It's more important to understand why we can manipulate equations to solve them.

A trinomial is just a polynomial with 3 terms. Like y = x^2 + x + 1. It doesn't even have to be a quadratic. It could be y = x^4 + x^3 + 1. It has three terms, so it's a trinomial.

The "perfect square" part is what is important. Just like a number can be a perfect square, a polynomial can be a perfect square too. For example, 25 is a perfect square since 5^2 = 25.

x^2 + 4x + 4 is also a perfect square because (x+2)^2 = x^2 + 4x + 4.
x^2 -2x + 1 is also a perfect square since it factors into (x-1)^2.

However, the polynomial x^2 - 5x + 6 is not a perfect square since it factors into (x-3)(x-2).

This is why I decided to factor x^2 + 8x + c into a "perfect square form" : (x+a)^2.

I can multiply (x+a)^2 to get: (x+a)(x+a) = x^2 + 2ax + a^2.

If I want this polynomial and your polynomial to be equal, I simply set them equal to each other.
x^2 + 2ax + a^2 = x^2 + 8x + c.

Subtracting x^2 from both sides: 2ax + a^2 = 8x + c.

Now you should see why a+a has to equal 8, and why a*a equals c.

Answer 6

Okay, I'm getting the gyst of it.. Could you help me with one more? Involving PST

Answer 7

Just for practice

Answer 8

PST?

Answer 9

What is the value of c so that x^2 +9x + c?

I got 81.. But I'm not sure.

PST (Perfect Square Trinomials)

Answer 10

I just assumed because 81 is a PST

Answer 11

Actually just PS lol

Answer 12

So we have x^2 + 9x + c. We want it to look like (x+a)^2.

If we multiply (x+a)^2 out, we get x^2 + 2ax + a^2.

We can set x^2 + 9x + c equal to x^2 + 2ax + a^2.

The x^2s go away on both sides. And we're left with:

9x + c = 2ax + a^2.

Answer 13

Polynomials are great because when you set them one equal to another, the coefficients on the x's must match up.

The coefficients are the numbers in front of x.

So 2a must equal 9. In other words, a = 9/2.

Answer 14

Thank you so much for your help. I love how you actually take the time to explain things thoroughly and so easily. I can tell you're going to be a huge help to many students on this site :)

Answer 15

We can also subtract everything from one to the other.
For example:

9x + c = 2ax + a^2

9x - 2ax + c - a^2 = 0

And then we can factor out an 'x'.

(9 - 2a)x + (c - a^2) = 0.

Now, what the previous equation says is that this polynomial is the zero polynomial.

So, we need all of the polynomial's coefficients to be equal to 0.

9 - 2a = 0 and c - a^2 = 0.

And it's easy to solve for 'a' from there.

Answer 16

And thank you! I should actually be studying for an exam of my own, but am procrastinating in my own way - haha.

Answer 17

Haha, well at least you're procrastinating in a good way :)

Answer 18

@Essin