Solve by completing the square: 2x^2 + 12x + 3 = 15

To solve the quadratic equation \( 2x^2 + 12x + 3 = 15 \) by completing the square, follow these steps:

### Step 1: Move the constant term to the other side
First, subtract 15 from both sides:
\[
2x^2 + 12x + 3 - 15 = 0
\]
\[
2x^2 + 12x - 12 = 0
\]
\[
2x^2 + 12x = 12
\]

### Step 2: Divide by the coefficient of \(x^2\)
Next, divide the entire equation by 2 to simplify:
\[
x^2 + 6x = 6
\]

### Step 3: Complete the square
To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides. The coefficient of \(x\) is 6, so:
\[
\left( \frac{6}{2} \right)^2 = 9
\]
Add 9 to both sides:
\[
x^2 + 6x + 9 = 6 + 9
\]
\[
(x + 3)^2 = 15
\]

### Step 4: Solve for \(x\)
Now, take the square root of both sides:
\[
x + 3 = \pm \sqrt{15}
\]
\[
x = -3 \pm \sqrt{15}
\]

Thus, the solutions are:
\[
x = -3 + \sqrt{15} \quad \text{or} \quad x = -3 - \sqrt{15}
\]

Would you like further details on any of the steps?

Here are 5 related questions you could explore:
1. How do you complete the square for a quadratic with a coefficient other than 1 in front of \(x^2\)?
2. What are other methods to solve quadratic equations apart from completing the square?
3. How do you find the vertex of a parabola using the completed square form?
4. How is completing the square related to the quadratic formula?
5. Can you always solve a quadratic equation using the method of completing the square?

**Tip:** Completing the square is a useful technique for converting a quadratic equation into a form that makes it easy to find the vertex or solve it.

Learn how to solve the quadratic equation 2x^2 + 12x + 3 = 15 using the completing the square method. This detailed step-by-step solution includes key algebra concepts, like transforming quadratic equations into perfect square form. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation