Question

How do you factor completely `a^2 - 10ab + 3b^2`?

Answer 1

`a^2-10ab+3b^2 = (a-(5+sqrt(22))b)(a-(5-sqrt(22))b)`

Explanation:

To factor this, complete the square then use the difference of squares identity, which can be written:

`A^2-B^2 = (A-B)(A+B)`

We use this with `A=(a-5b)` and `B=sqrt(22)b` as follows:

`a^2-10ab+3b^2 = a^2-10ab+25b^2-22b^2`

`color(white)(a^2-10ab+3b^2) = a^2-2a(5b)+(5b)^2-22b^2`

`color(white)(a^2-10ab+3b^2) = (a-5b)^2-(sqrt(22)b)^2`

`color(white)(a^2-10ab+3b^2) = ((a-5b)-sqrt(22)b)((a-5b)+sqrt(22)b)`

`color(white)(a^2-10ab+3b^2) = (a-(5+sqrt(22))b)(a-(5-sqrt(22))b)`