https://file.aiquickdraw.com/mathbot/file/1725515063797mw3lrdun.jpg

To solve the quadratic equation \(-3x^2 + 12x + 11 = 0\) by completing the square, we will first make sure the quadratic coefficient is 1 by dividing the entire equation by \(-3\).

### Step-by-step process:

1. **Divide by -3** to simplify the quadratic term:
   \[
   x^2 - 4x - \frac{11}{3} = 0
   \]

2. **Move the constant term** to the other side:
   \[
   x^2 - 4x = \frac{11}{3}
   \]

3. **Complete the square**:
   - Take half of the coefficient of \(x\) (which is \(-4\)), square it, and add it to both sides.
   - Half of \(-4\) is \(-2\), and \((-2)^2 = 4\).

   Add 4 to both sides:
   \[
   x^2 - 4x + 4 = \frac{11}{3} + 4
   \]
   \[
   (x - 2)^2 = \frac{11}{3} + \frac{12}{3} = \frac{23}{3}
   \]

4. **Solve for \(x\)** by taking the square root of both sides:
   \[
   x - 2 = \pm \sqrt{\frac{23}{3}}
   \]

5. **Isolate \(x\)**:
   \[
   x = 2 \pm \frac{\sqrt{23}}{\sqrt{3}} = 2 \pm \frac{\sqrt{69}}{3}
   \]

Thus, the solutions are:
\[
x = 2 + \frac{\sqrt{69}}{3} \quad \text{or} \quad x = 2 - \frac{\sqrt{69}}{3}
\]

Would you like more details on any step? Here are some related questions you can explore:

1. How do you simplify square roots with fractions?
2. What is the geometric interpretation of completing the square?
3. How is completing the square useful in solving quadratic equations?
4. Can you apply the quadratic formula to check the solution?
5. What are the conditions for a quadratic equation to have real solutions?

**Tip:** When completing the square, always ensure the coefficient of the quadratic term is 1 before proceeding.

-3x^2 + 12x + 11 = 0. Solve by completing the square.

Step-by-step guide on how to solve the quadratic equation -3x^2 + 12x + 11 = 0 using the completing the square method. Explore algebraic concepts and techniques suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation