Solve by completing the square.

w2 − 2w = 49

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

To solve the quadratic equation \( w^2 - 2w = 49 \) by completing the square, follow these steps:

### Step 1: Move the constant term to the right side of the equation.
\[
w^2 - 2w - 49 = 0
\]
\[
w^2 - 2w = 49
\]

### Step 2: Complete the square on the left side of the equation.
- To complete the square, take the coefficient of \( w \) (which is \(-2\)), divide it by 2, and then square the result.
\[
\left(\frac{-2}{2}\right)^2 = 1
\]
- Add 1 to both sides of the equation to maintain equality.
\[
w^2 - 2w + 1 = 49 + 1
\]
\[
(w - 1)^2 = 50
\]

### Step 3: Take the square root of both sides of the equation.
\[
w - 1 = \pm \sqrt{50}
\]

### Step 4: Solve for \( w \).
- Simplify the square root \( \sqrt{50} \):
\[
\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}
\]
- Therefore, the solutions are:
\[
w - 1 = \pm 5\sqrt{2}
\]
\[
w = 1 \pm 5\sqrt{2}
\]
- Expressing the solutions in decimal form:
\[
w_1 = 1 + 5\sqrt{2} \approx 1 + 7.07 = 8.07
\]
\[
w_2 = 1 - 5\sqrt{2} \approx 1 - 7.07 = -6.07
\]

### Final Answer:
The solutions are:
\[
w \approx 8.07 \quad \text{or} \quad w \approx -6.07
\]

Would you like any details on this process or have any other questions?

Here are 5 questions you might consider next:
1. How can you check the solutions of a quadratic equation?
2. What is the significance of the discriminant in quadratic equations?
3. Can completing the square be used for any quadratic equation?
4. What are the steps to solve a quadratic equation by factoring?
5. How does completing the square relate to the quadratic formula?

**Tip:** Always verify your solutions by substituting them back into the original equation.

Learn how to solve the quadratic equation w^2 - 2w = 49 using the method of completing the square. This detailed step-by-step solution covers key algebraic concepts such as quadratic equations, completing the square, and square roots. Suitable for high school students (Grades 10-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation