Sure, let's simplify the given expression [tex]\( 2m^2 - 2mn + n^2 - 9r \)[/tex].
### Step-by-Step Solution:
1. Identify the expression:
[tex]\[ 2m^2 - 2mn + n^2 - 9r \][/tex]
2. Group like terms where possible:
In this expression, we have terms involving [tex]\( m \)[/tex], terms involving [tex]\( n \)[/tex], and a term involving [tex]\( r \)[/tex]. However, given the structure, not all terms can be directly combined because they involve different variables.
3. Look for patterns and factor where applicable:
Let's take a close look at the first three terms: [tex]\( 2m^2 - 2mn + n^2 \)[/tex]. These terms can form a perfect square trinomial.
To recognize a perfect square trinomial, we check if it can be written in the form:
[tex]\[ (am + bn)^2 \][/tex]
Expanding [tex]\( (am + bn)^2 \)[/tex] gives:
[tex]\[ a^2m^2 + 2abmn + b^2n^2 \][/tex]
Comparing [tex]\( 2m^2 - 2mn + n^2 \)[/tex] with [tex]\( a^2m^2 + 2abmn + b^2n^2 \)[/tex]:
- [tex]\( a^2m^2 \)[/tex] corresponds to [tex]\( 2m^2 \)[/tex], thus [tex]\( a = \sqrt{2} \)[/tex].
- [tex]\( b^2n^2 \)[/tex] corresponds to [tex]\( n^2 \)[/tex], thus [tex]\( b = 1 \)[/tex].
- For [tex]\( 2abmn \)[/tex] to match [tex]\( -2mn \)[/tex], we should have [tex]\( 2(\sqrt{2})(1) = 2\sqrt{2} \)[/tex]. However, our expression contains [tex]\(-2mn\)[/tex], indicating no direct matching using simple integers.
Therefore, the expression [tex]\( 2m^2 - 2mn + n^2 \)[/tex] does not factor neatly into a perfect square.
4. Re-examine simplification:
Since the terms involving [tex]\( m \)[/tex] and [tex]\( n \)[/tex] do not simplify further, and no common factors exist across all terms, the expression remains as it is.
Thus, the simplified form of the expression [tex]\( 2m^2 - 2mn + n^2 - 9r \)[/tex] is:
[tex]\[ 2m^2 - 2mn + n^2 - 9r \][/tex]
There are no further simplifications possible with the given terms and their distinct variables. Hence, the expression in its simplest form is:
[tex]\[ 2m^2 - 2mn + n^2 - 9r \][/tex]
