Understanding And Solving Quadratic Equations: X^2 + 2X + 6 = 0

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Introduction

When it comes to algebra, quadratic equations can be a bit challenging. They are equations that involve a variable raised to the power of two. One common quadratic equation is x^2 + 2x + 6 = 0. In this article, we will break down the steps to solve this equation.

Step 1: Identify the Values of a, b, and c

The quadratic equation is in the form of ax^2 + bx + c = 0. In our equation, a = 1, b = 2, and c = 6. It is important to identify these values before proceeding to the next step.

Step 2: Use the Quadratic Formula

The quadratic formula is the most common way to solve quadratic equations. The formula is x = (-b ± √(b^2 - 4ac)) / 2a. Substituting the values of a, b, and c from our equation, we get x = (-2 ± √(-20)) / 2. Since the square root of a negative number is not a real number, we cannot proceed with this method.

Step 3: Use Completing the Square

Completing the square is another method to solve quadratic equations. By adding and subtracting a constant, we can change the equation to a perfect square trinomial. The equation can then be solved by taking the square root of both sides. To use this method, we first need to isolate the x^2 term. Subtracting 6 from both sides, we get x^2 + 2x = -6. We then add (2/2)^2 = 1 to both sides, which gives us x^2 + 2x + 1 = -5 + 1. Simplifying further, we get (x+1)^2 = -4. We can then take the square root of both sides, which gives us x+1 = ±2i. Solving for x, we get x = -1 ±2i.

Step 4: Check Your Solution

It is always important to check your solution to ensure that it satisfies the original equation. Substituting x = -1 + 2i, we get (-1 + 2i)^2 + 2(-1 + 2i) + 6 = 0. Simplifying the equation, we get -4 + 4i = 0, which is not true. Therefore, our solution is not valid.

Step 5: Final Solution

We can also solve the equation by factoring. Since the coefficient of x^2 is 1, we can easily factor the equation to (x + 3i)(x - 3i) = 0. This gives us two solutions: x = 3i and x = -3i.

Conclusion

In conclusion, solving quadratic equations can be challenging but with the right steps, it can be done easily. By identifying the values of a, b, and c, we can use the quadratic formula to solve the equation. However, if the square root of a negative number is obtained, we can use completing the square or factoring methods to solve the equation. Always remember to check your solution to ensure that it satisfies the original equation.