Question

How do you factor `x^2 - 45x + 450`?

Answer 1

`x^2-45x+450 = (x-30)(x-15)`

Explanation:

Note that `45=30+15` and `450 = 30*15`

Hence:

`x^2-45x+450 = (x-30)(x-15)`

Alternatively, you could find it by completing the square:

`4(x^2-45x+450) = 4x^2-180x+1800`

`color(white)(4(x^2-45x+450)) = (2x-45)^2-45^2+1800`

`color(white)(4(x^2-45x+450)) = (2x-45)^2-2025+1800`

`color(white)(4(x^2-45x+450)) = (2x-45)^2-225`

`color(white)(4(x^2-45x+450)) = (2x-45)^2-15^2`

`color(white)(4(x^2-45x+450)) = ((2x-45)-15)((2x-45)+15)`

`color(white)(4(x^2-45x+450)) = (2x-60)(2x-30)`

`color(white)(4(x^2-45x+450)) = 4(x-30)(x-15)`

So dividing both ends by `4` we find:

`x^2-45x+450 = (x-30)(x-15)`