What is the factored form of q 2 12q 36

Complete step-by-step solution:
We need to factorize the given expression. We know that the given expression is a polynomial in the variable $q$ . Also, the exponent of the highest degree term in a polynomial is known as its degree. Therefore, we can see that the degree of the given polynomial is $2$ .
The polynomial of degree $2$ is called a quadratic polynomial.
Now, let us take the given expression as
$f(q) = {q^2} - 12q + 36$ .
We represented the given expression as $f(q)$ . Also, the given expression is a quadratic polynomial.
Now, to factorize $f(q)$ , we have to find two numbers $a$ and $b$such that $a + b = - 12$ and $ab = 36$ . So clearly, $- 6 - 6 = - 12$ and $- 6 \times - 6 = 36$ .
So, we write the middle term $- 12q$ as $- 6q - 6q$
∴ ${q^2} - 12q + 36$
$= {q^2} - 6q - 6q + 36$
Now. from the first two terms we took out $q$ as common and from the next two terms we took out $- 6$ as common.
$= q(q - 6) - 6(q - 6)$
$= (q - 6)(q - 6)$ , now taking out as common $(q - 6)$ from both the terms

${q^2} - 12q + 36 = (q - 6)(q - 6)$ , which is the required factored form.

Note: The given expression is a quadratic polynomial because the degree of the given polynomial is $2$ . The name ‘quadratic’ has been derived from ‘quadrate’, which means ‘square’. Generally, any quadratic polynomial in variable $x$ with real coefficients is of the form $f(x) = a{x^2} + bx + c$ , where $a$ , $b$ and $c$ are real numbers and $a \ne 0$ .

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