Get expert advice and community support for your questions on IDNLearn.com. Join our community to receive prompt, thorough responses from knowledgeable experts.

Fill in the gaps in the equation below by completing the square.

[tex]\[ x^2 - 6x - 1 = (x - \square)^2 - \square \][/tex]


Sagot :

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]