Factor the following expression: $81m^2-108mn+36n^2$
Understand the Problem
The prompt is asking to simplify the quadratic expression. We need to factor the given quadratic expression, $81m^2 - 108mn + 36n^2$, to its simplest form by identifying common factors and applying algebraic identities.
Answer
$9(3m - 2n)^2$
Answer for screen readers
$9(3m - 2n)^2$
Steps to Solve
- Identify the common factor
First, we look for the greatest common factor (GCF) of the coefficients in the quadratic expression $81m^2 - 108mn + 36n^2$. The GCF of 81, 108, and 36 is 9. Factoring out 9 from the expression gives:
$$9(9m^2 - 12mn + 4n^2)$$
- Recognize the perfect square trinomial
Next, we examine the expression inside the parentheses, $9m^2 - 12mn + 4n^2$. We can rewrite it as $(3m)^2 - 2(3m)(2n) + (2n)^2$. This is a perfect square trinomial of the form $a^2 - 2ab + b^2$, where $a = 3m$ and $b = 2n$.
- Factor the perfect square trinomial
We know that $a^2 - 2ab + b^2 = (a - b)^2$. Substituting $a = 3m$ and $b = 2n$, we get:
$$(3m - 2n)^2$$
- Write the fully factored expression
Now, we substitute this back into the expression from step 1:
$$9(3m - 2n)^2$$
Since $9 = 3^2$, we can rewrite the expression as:
$$3^2(3m - 2n)^2 = [3(3m - 2n)]^2 = (9m - 6n)^2$$
However, we want the simplest factored form, so we should stick to $9(3m-2n)^2$ or $3^2(3m-2n)^2$
$9(3m - 2n)^2$
More Information
The simplified form of the given quadratic expression is $9(3m-2n)^2$. This represents a perfect square, where $3m - 2n$ squared, multiplied by 9 yields the original expression.
Tips
A common mistake would be not recognizing the perfect square trinomial and attempting to factor it using other methods, which can be more complex and error-prone. Another mistake could be not factoring out the greatest common factor in the first step, leading to larger numbers and more complicated factoring later on. Also, forgetting to include the "9" after factoring the perfect square is a common mistake.
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