Sure, let's complete the square for the equation [tex]\( x^2 - 6x - 1 \)[/tex].
1. Start with the given quadratic expression on the left side:
[tex]\[
x^2 - 6x - 1
\][/tex]
2. To complete the square, we focus on the [tex]\( x^2 - 6x \)[/tex] part. We need to transform this expression into a perfect square trinomial. This is done by taking the coefficient of [tex]\( x \)[/tex] (which is -6), dividing it by 2, and then squaring the result:
[tex]\[
\left( \frac{-6}{2} \right)^2 = (-3)^2 = 9
\][/tex]
3. Now, rewrite the quadratic expression by adding and subtracting this square value (9):
[tex]\[
x^2 - 6x + 9 - 9 - 1 = (x - 3)^2 - 9 - 1
\][/tex]
4. Combine the constants:
[tex]\[
(x - 3)^2 - 10
\][/tex]
Therefore, the completed square form of the given equation is:
[tex]\[
x^2 - 6x - 1 = (x - \square)^2 - \square
\][/tex]
Fill in the blanks:
[tex]\[
x^2 - 6x - 1 = (x - 3)^2 - 10
\][/tex]
So, the final answer is:
[tex]\[
x^2 - 6x - 1 = (x - 3)^2 - 10
\][/tex]
Thus:
[tex]\[
\square = 3 \quad \text{and} \quad \square = 10
\][/tex]
