Question

Leia uses the method of completing the square to transform the quadratic equation
x
2
−
6
x
+
1
=
0
into an equivalent equation of the form
(
x
−
3
)
2
=
□
.

What is the missing number in Leia's equivalent equation? Enter the missing number in the box below.

(
x
−
3
)
2
=

Answer 1

To complete the square for the quadratic equation \(x^2 - 6x + 1 = 0\), Leia aims to rewrite it in the form \((x - 3)^2 = \square\).

Here are the steps to find the missing number in Leia's equivalent equation:

1. Begin with the quadratic equation: \(x^2 - 6x + 1 = 0\).

2. To complete the square, focus on the coefficient of the x-term, which is -6.

3. Take half of the coefficient of x, square it, and add it to both sides of the equation:

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

\[x^2 - 6x + 9 = -1 + 9\]

\[x^2 - 6x + 9 = 8\]

4. Rewrite the left side of the equation as a squared binomial:

\[(x - 3)^2 = 8\]

Therefore, the missing number in Leia's equivalent equation is 8, making the equation \((x - 3)^2 = 8\). This form highlights the completed square expression for the given quadratic equation.