Question

How do you factor the expression `x^2 - 12x + 36`?

Answer 1

By using factorization algorithm, we can factor any given expression.

Explanation:

For quadratic polynomials, the algorithm is as follows:
First, multiply the coefficient of the highest degree term and the constant. In this case, it is (1).(36)=36
Now, check the factors of the product and find how many different ways they can be arranged to get the product.
36=1.36
=2.18
=4.9
=6.6
=12.3

Now, you have to choose the pair of factors in such a way that adding them or subtracting them must be equal to the middle term coefficient.
We choose 6.6 because -6-6=-12 which is the coefficient of the middle term.
Now, split the middle term as -6x-6x, since the factors we chose are -6 and -6.
That is, `x^2` -6x-6x+36.
Now, take out the common factors from each pair.
That is, x(x-6)-6(x-6)
Finally, (x-6)(x-6) is the required factored form.

Answer 2

`(x-6)^2`

Explanation:

`color(blue)((1))x^2-color(red)(12x)+36=x^2-color(red)(6x-6x)+36`
`=color(red)(x)(x-6)-color(red)(6)(x-6)`
`=(x-6)(x-6)`
`=(x-6)^2`
OR,Using `color(red)(A^2-2AB+B^2=(A-B)^2)`
We get,
`x^2-12x+36=(x)^2-2(x)(6)+(6)^2=(x-6)^2`