Complete the square of the given function.
f(x) = x^2 - 6x
HINT GIVEN:
If f(x) = x^2 - 2x, then f(x) = (x^2 - 2x + 1) - 1 = (x - 1)^2 - 1.
I am looking for steps.
Sample:
Step 1: do this
Step 2: do that, etc.
If you have an expression, say
$\displaystyle x^2 + 3x$
that you need to complete the square of, consider the following:
$\displaystyle x^2 + 2ax + a^2 = (x + a)^2$
So we wish to add a number to our expression to put it in the above form.
So equate the coefficients of the linear terms:
$\displaystyle 3 = 2a$
Thus
$\displaystyle a = \frac{3}{2}$
According to the prescription, then, we need to add an $\displaystyle a^2$ to our $\displaystyle x^2 + 3x$ to make it a perfect square. But what we add on one side, we must also subtract (so we're adding a net of zero to that side), thus:
$\displaystyle x^2 + 3x = x^2 + 3x + \left ( \frac{3}{2} \right ) ^2 - \left ( \frac{3}{2} \right ) ^2$
$\displaystyle = \left ( x + \frac{3}{2} \right ) ^2 - \left ( \frac{3}{2} \right ) ^2$
-Dan