Solve each equation by completing the square: xsqrd-12x+27=0

To solve the quadratic equation \(x^2 - 12x + 27 = 0\) by completing the square, follow these steps:

### Step 1: Isolate the constant term
Rewrite the equation so that the constant term is on the other side:
\[
x^2 - 12x = -27
\]

### Step 2: Complete the square
Take half the coefficient of \(x\), square it, and add it to both sides of the equation. 

The coefficient of \(x\) is \(-12\), so:
\[
\left(\frac{-12}{2}\right)^2 = 36
\]

Add \(36\) to both sides:
\[
x^2 - 12x + 36 = -27 + 36
\]

Simplify:
\[
(x - 6)^2 = 9
\]

### Step 3: Solve for \(x\)
Take the square root of both sides:
\[
x - 6 = \pm \sqrt{9}
\]
\[
x - 6 = \pm 3
\]

### Step 4: Solve for \(x\) explicitly
Separate into two cases:
1. \(x - 6 = 3 \implies x = 9\)
2. \(x - 6 = -3 \implies x = 3\)

### Final Answer:
\[
x = 9 \quad \text{or} \quad x = 3
\]

Would you like a step-by-step breakdown for verification or more examples?  

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### Related Questions:
1. What is completing the square, and why does it work?
2. How can completing the square be used to rewrite a quadratic in vertex form?
3. Can every quadratic equation be solved using completing the square? Why or why not?
4. How does completing the square compare to the quadratic formula?
5. What are some real-world applications of solving quadratic equations?

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**Tip:** Completing the square is also helpful in graphing quadratics, as it directly gives the vertex of the parabola.

Solve each equation by completing the square: x^2 - 12x + 27 = 0

Learn how to solve the quadratic equation x^2 - 12x + 27 = 0 using the completing the square method. This detailed step-by-step solution covers core algebraic concepts and explains how to rewrite the equation for easier solving. Suitable for Grades 8-10 students.

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