Factor by completing the square for` f(x)= x^2 -4x +9`
First, when we complete the square for a quadratic function of the form `f(x)=ax^2+bx+c` we get it into the form `(x+b/2)^2-(b/2)^2+c` and then simplify. To do this we must add and subtract (to not change the equation) factors of `(b/2)^2` .
In this case:
Now add and subtract factors of `(b/2)^2`
`x^2 -4x +9+(4-4)`
Factor the term in the parenthesis.
This is now in the form ` (x+b/2)^2-(b/2)^2+c` , now simplify.
This form is convenient for reading off the horizontal and vertical shift of the parabola. The vertex is at `(2,5)` .
Express `f(x) = x^2 - 4x + 9` as `x^2 - 2*x*2 + 2^2 - 2^2 + 9 = (x - 2)^2 - 4 + 9 = (x - 2)^2 + 5.` We used the formula `(a - b)^2 = a^2 - 2ab + b^2` in the reverse direction.
We see that this function is always positive for real `x,` therefore it cannot be factored using real coefficients.
But it can using complex numbers: `-5 = (i sqrt(5))^2` and
`f(x) =(x - 2)^2 + 5 =(x - 2)^2 -(i sqrt(5))^2 = (x - 2 -i sqrt(5))(x - 2 + i sqrt(5)).`
Here we used the formula `a^2 - b^2 = (a - b)(a + b).`
The answer is impossible for real coefficients and `(x - 2 -i sqrt(5))(x - 2 + i sqrt(5))` for complex coefficients.