Solve for the roots in *simplest form* by completing the square:

2, x, squared, plus, 36, x, plus, 34, equals, 0

2*x*2+36*x*+34=

0

We are given the quadratic equation:

\[
2x^2 + 36x + 34 = 0
\]

### Step 1: Divide by the coefficient of \(x^2\)
To complete the square, first divide the entire equation by 2 to simplify the leading coefficient of \(x^2\):

\[
x^2 + 18x + 17 = 0
\]

### Step 2: Move the constant to the other side
Subtract 17 from both sides to isolate the terms involving \(x\):

\[
x^2 + 18x = -17
\]

### Step 3: Complete the square
To complete the square, take half of the coefficient of \(x\) (which is 18), square it, and add it to both sides of the equation:

\[
\left(\frac{18}{2}\right)^2 = 9^2 = 81
\]

Now add 81 to both sides:

\[
x^2 + 18x + 81 = -17 + 81
\]

\[
x^2 + 18x + 81 = 64
\]

### Step 4: Write the left side as a square
The left-hand side is now a perfect square trinomial, which can be factored as:

\[
(x + 9)^2 = 64
\]

### Step 5: Solve for \(x\)
Take the square root of both sides:

\[
x + 9 = \pm \sqrt{64}
\]

\[
x + 9 = \pm 8
\]

### Step 6: Isolate \(x\)
Now solve for \(x\) by subtracting 9 from both sides:

1. \(x + 9 = 8\) gives \(x = 8 - 9 = -1\)
2. \(x + 9 = -8\) gives \(x = -8 - 9 = -17\)

### Final Answer:
The roots of the equation are:

\[
x = -1 \quad \text{or} \quad x = -17
\]

Would you like further clarification on any step?

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Here are 5 related questions:
1. How do you determine when to use completing the square?
2. What is the quadratic formula and how does it compare to completing the square?
3. Can all quadratic equations be solved using completing the square?
4. What are the steps to convert a quadratic equation from standard form to vertex form using completing the square?
5. How does completing the square relate to solving quadratic inequalities?

**Tip**: Completing the square is especially useful when the quadratic equation does not factor nicely or when transforming the quadratic into a different form, such as vertex form.

Solve for the roots in simplest form by completing the square: 2x^2 + 36x + 34 = 0

Learn how to solve the quadratic equation 2x^2 + 36x + 34 by completing the square with a detailed step-by-step explanation. This problem highlights key algebraic methods, including simplifying the equation and finding its roots. Perfect for students in Grades 9-10.

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