To solve the quadratic equation $3x^2 + 12x - 6 = 0$ by completing the square, follow these steps:

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Step 1: Simplify by dividing through by the leading coefficient (if not 1)

Divide through by $3$ to make the coefficient of $x^2$ equal to 1: $x^2 + 4x - 2 = 0$

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Step 2: Isolate the constant term

Move the constant term ($-2$) to the other side of the equation: $x^2 + 4x = 2$

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Step 3: Complete the square

To complete the square, take half of the coefficient of $x$ ($4$), square it, and add it to both sides: $\text{Half of } 4 = 2,\quad (2)^2 = 4$ Add $4$ to both sides: $x^2 + 4x + 4 = 2 + 4$ $(x + 2)^2 = 6$

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Step 4: Solve for $x$

Take the square root of both sides: $x + 2 = \pm\sqrt{6}$ Subtract $2$ from both sides: $x = -2 \pm \sqrt{6}$

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Final Answer:

$x = -2 + \sqrt{6} \quad \text{or} \quad x = -2 - \sqrt{6}$

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Would you like more details or have any questions? 😊

Relative Questions:

1. How is completing the square related to the quadratic formula?
2. What are other methods to solve a quadratic equation?
3. Can completing the square be used to derive the vertex form of a quadratic equation?
4. Why do we divide by the leading coefficient in the first step?
5. What happens when the discriminant ($b^2 - 4ac$) is negative?

Tip:

When completing the square, always ensure the coefficient of $x^2$ is 1 before proceeding.