Solve the following quadratic equation by completing the square.


𝑥2−12𝑥=−1*x*2−12*x*=−1​​

Enter multiple answers separated by a comma or enter NoneNone​ if there are no real solutions.


𝑥=*x*=


To solve the quadratic equation \( x^2 - 12x = -1 \) by completing the square, follow these steps:

### Step 1: Move the constant term to the other side
First, add \( 1 \) to both sides of the equation to isolate the quadratic and linear terms:
\[
x^2 - 12x + \_ = 1 + \_
\]

### Step 2: Complete the square
To complete the square, take half of the coefficient of \( x \), which is \(-12\), divide by 2, and then square the result:
\[
\left(\frac{-12}{2}\right)^2 = (-6)^2 = 36
\]

Add \( 36 \) to both sides of the equation:
\[
x^2 - 12x + 36 = 1 + 36
\]
\[
x^2 - 12x + 36 = 37
\]

### Step 3: Write the left side as a squared binomial
The left side of the equation can now be written as a perfect square trinomial:
\[
(x - 6)^2 = 37
\]

### Step 4: Solve for \( x \)
Take the square root of both sides:
\[
x - 6 = \pm\sqrt{37}
\]
\[
x = 6 \pm \sqrt{37}
\]

### Final Answer:
The solutions are:
\[
x = 6 + \sqrt{37}, \quad x = 6 - \sqrt{37}
\]

So, \( x = 6 + \sqrt{37}, 6 - \sqrt{37} \).

Would you like further details or have any questions?

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Here are 5 related questions to explore:

1. How would the solutions change if the equation was \( x^2 - 12x = 0 \)?
2. What is the significance of the discriminant in determining the number of real solutions?
3. How can completing the square be used to find the vertex of a parabola?
4. Can you solve \( x^2 - 12x + 20 = 0 \) by completing the square?
5. How does the method of completing the square compare to using the quadratic formula?

**Tip:** When solving quadratic equations, always check if completing the square is more straightforward than using the quadratic formula, especially when the equation can be easily manipulated into a perfect square trinomial.

Learn how to solve the quadratic equation x^2 - 12x = -1 by completing the square with a detailed step-by-step solution. This problem involves completing the square technique and quadratic formula, suitable for students in Grades 10-12.

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