Solve by completing the square.

–2s2 − 20s = –42

Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

We are tasked with solving the quadratic equation \( -2s^2 - 20s = -42 \) by completing the square. Let’s break it down step by step.

### Step 1: Move the constant term to the other side
To begin, we'll add \(42\) to both sides to move the constant term to the right-hand side of the equation:

\[
-2s^2 - 20s + 42 = 0
\]

Next, rewrite the equation as:

\[
-2s^2 - 20s = 42
\]

### Step 2: Divide by the coefficient of \(s^2\)
To make the coefficient of \(s^2\) equal to 1, divide the entire equation by \(-2\):

\[
s^2 + 10s = -21
\]

### Step 3: Complete the square
To complete the square, take half of the coefficient of \(s\) (which is \(10\)), square it, and add it to both sides of the equation:

\[
\left(\frac{10}{2}\right)^2 = 25
\]

Now, add \(25\) to both sides:

\[
s^2 + 10s + 25 = -21 + 25
\]

Simplifying:

\[
s^2 + 10s + 25 = 4
\]

### Step 4: Factor the left-hand side
The left-hand side is a perfect square trinomial, so it factors as:

\[
(s + 5)^2 = 4
\]

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

\[
s + 5 = \pm 2
\]

Now, solve for \(s\) by isolating \(s\):

\[
s = -5 \pm 2
\]

### Step 6: Find the solutions
There are two possible solutions:

1. \( s = -5 + 2 = -3 \)
2. \( s = -5 - 2 = -7 \)

### Final Answer:
The solutions are \( s = -3 \) and \( s = -7 \).

Would you like any further clarification?

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Here are 5 additional questions related to completing the square:

1. How can you apply completing the square to solve quadratic equations with fractional coefficients?
2. What is the geometric interpretation of completing the square?
3. How does completing the square relate to the quadratic formula?
4. Can completing the square be used to derive the quadratic formula?
5. How do you handle completing the square when the equation has imaginary solutions?

**Tip:** Always double-check your square terms when completing the square to avoid mistakes with coefficients.

Solve by completing the square. –2s^2 − 20s = –42. Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

Solve the quadratic equation -2s^2 − 20s = -42 using the method of completing the square. This step-by-step guide explores algebraic techniques, including factoring, perfect square trinomials, and the square root method. Suitable for Grades 9-11.

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