How do you solve 3x² + 6x - 2 = 0 by completing the square?

To solve the equation 3x² + 6x - 2 = 0 by completing the square, follow these steps:

Step 1: Rearrange the Equation

First, move the constant term to the other side:

3x² + 6x = 2

Step 2: Factor Out the Coefficient of x²

Next, factor out the 3 from the left side:

3(x² + 2x) = 2

Step 3: Complete the Square

To complete the square, take half of the coefficient of x (which is 2), square it, and add it inside the parentheses:

• Half of 2 is 1.
• Squaring 1 gives you 1.

Add and subtract this square inside the parentheses:

3(x² + 2x + 1 - 1) = 2

This simplifies to:

3((x + 1)² - 1) = 2

Step 4: Simplify the Equation

Distribute the 3:

3(x + 1)² - 3 = 2

Now, add 3 to both sides:

3(x + 1)² = 5

Step 5: Isolate the Square

Divide both sides by 3:

(x + 1)² = \frac{5}{3}

Step 6: Take the Square Root

Now, take the square root of both sides:

x + 1 = ±√\frac{5}{3}

Step 7: Solve for x

Finally, isolate x:

x = -1 ± √\frac{5}{3}

Final Answers

The solutions to the equation are:

• x = -1 + √\frac{5}{3}
• x = -1 - √\frac{5}{3}