Question

Write x² – 6x + 1 in the form (x + a)² + b where a and b are integers.

Answer 1

The quadratic equation x² - 6x + 1 can be rewritten as (x - 3)² - 8 by completing the square, where 'a' is -3 and 'b' is -8.

Step-by-step explanation:

The student has asked to write the quadratic equation x² − 6x + 1 in the form (x + a)² + b, where 'a' and 'b' are integers. To complete the square for the quadratic term and the linear term, we take half of the coefficient of the x term, which is -6/2, giving us -3, and then square it to get 9. Adding and subtracting 9 inside the parentheses allows us to create a perfect square trinomial.

The steps are as follows:

1. Start with the original quadratic equation: x² − 6x + 1.
2. Add and subtract 9 inside the equation: x² − 6x + 9 - 9 + 1.
3. Rewrite the equation grouping the perfect square trinomial: (x² − 6x + 9) - 8.
4. Factor the perfect square trinomial: (x - 3)² - 8.

Hence, the quadratic equation x² - 6x + 1 can be written in the form (x - 3)² - 8.

Answer 2

To express the quadratic expression x² - 6x + 1 in the form (x + a)² + b, you need to complete the square. Here's how you can do it:

Start with the given expression: x² - 6x + 1.

To complete the square, you need to add and subtract a constant term inside the parentheses in such a way that the expression remains equivalent:

x² - 6x + 1 = (x² - 6x + 9 - 9) + 1

Notice that we added 9 and subtracted 9 inside the parentheses.

Now, group the perfect square trinomial (x² - 6x + 9) and the constant term (-9 + 1) separately:

(x² - 6x + 9) - 9 + 1

Simplify each group:

(x - 3)² - 8

Now, the expression is in the desired form (x + a)² + b, where:

a = -3

b = -8

So, x² - 6x + 1 can be expressed as (x - 3)² - 8.

Step-by-step explanation: