QuestionFactor completely. $c^{2}+25-10 c$

Studdy Solution

STEP 1

What is this asking? We need to rewrite this expression as a product of simpler expressions. Watch out! Don't forget to rearrange the terms to make it easier to factor!

STEP 2

1. Rearrange Terms
2. Recognize the Pattern
3. Factor the Expression

STEP 3

Let's rearrange the terms in the expression $c^{2}+25-10c$ to make it easier to see any patterns.
Notice that we have a $c^2$ term, a $c$ term, and a constant term.
Let's write it in the standard form of a quadratic expression, which is $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants.

STEP 4

By simply reordering the terms, we get $c^{2} - 10c + 25$ .
This looks much better!

STEP 5

Now, let's see if we can spot any patterns.
Does this rearranged expression remind us of anything?
It looks like a perfect square trinomial!
A perfect square trinomial is of the form $a^2 \pm 2ab + b^2$ , which factors into $(a \pm b)^2$ .

STEP 6

In our case, we have $c^2 - 10c + 25$ .
If we think of $c$ as our $a$ , then $a^2$ is $c^2$ .
What about $b$ ?
If $b^2 = 25$ , then $b$ must be $5$ .
Now, is the middle term, $-10c$ , equal to $-2ab$ ?
Let's check: $-2 \cdot c \cdot 5 = -10c$ .
It matches perfectly!

STEP 7

Since we've confirmed that our expression is a perfect square trinomial, we can factor it!
Remember, $a^2 - 2ab + b^2$ factors into $(a-b)^2$ .

STEP 8

In our case, $a = c$ and $b = 5$ , so $c^2 - 10c + 25$ factors into $(c-5)^2$ .
And that's it!
We've completely factored the expression.

STEP 9

The completely factored form of $c^2 + 25 - 10c$ is $(c-5)^2$ .

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