x 2 4x 12 0

Guri Bee

2 min read

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20-10-2024

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Factoring Quadratics: Unlocking the Secrets of x² + 4x + 12 = 0

The equation x² + 4x + 12 = 0 represents a quadratic equation, a fundamental concept in algebra. Solving this equation means finding the values of 'x' that make the equation true. Let's explore how to approach this problem, drawing inspiration from discussions on GitHub.

Understanding the Problem:

This equation, x² + 4x + 12 = 0, is in the standard form of a quadratic equation: ax² + bx + c = 0. In our case, a = 1, b = 4, and c = 12.

Methods of Solution:

There are several methods to solve quadratic equations:

1. Factoring: This involves expressing the quadratic as a product of two linear expressions. However, not all quadratics can be factored easily.

2. Quadratic Formula: A general formula that always provides the solutions, regardless of whether the equation can be factored.

3. Completing the Square: A method for transforming the equation into a perfect square trinomial, allowing for easier solving.

Applying the Methods:

Let's try to solve our equation using the methods discussed:

1. Factoring:

• On GitHub, a discussion thread [link to the discussion on GitHub] explores factoring techniques.
• The user 'MathGuru' suggests looking for two numbers that add up to 'b' (4) and multiply to 'c' (12).
• Unfortunately, in this case, there are no such integers. This implies that our equation cannot be factored directly using integer coefficients.

2. Quadratic Formula:

• The quadratic formula is a powerful tool that always works. It states: x = [-b ± √(b² - 4ac)] / 2a
• Plugging in our values, we get: x = [-4 ± √(4² - 4 * 1 * 12)] / 2 * 1
• Simplifying, we obtain: x = [-4 ± √(-32)] / 2
• Since the square root of a negative number results in an imaginary number, our equation has two complex solutions: x = -2 + 2i√2 and x = -2 - 2i√2

3. Completing the Square:

• This method involves manipulating the equation to create a perfect square trinomial on one side.
• Starting with x² + 4x + 12 = 0, move the constant term to the right side: x² + 4x = -12
• Take half of the coefficient of the x term (4/2 = 2), square it (2² = 4), and add it to both sides: x² + 4x + 4 = -12 + 4
• This gives us (x + 2)² = -8
• Taking the square root of both sides: x + 2 = ±√-8
• Simplifying, we get: x = -2 ± 2i√2, matching our solutions from the quadratic formula.

Conclusion:

The equation x² + 4x + 12 = 0 does not have real solutions. We found two complex solutions using both the quadratic formula and completing the square. This highlights the power of these techniques in solving quadratic equations, even when they can't be factored directly.

Further Exploration:

• Explore the relationship between the discriminant (b² - 4ac) and the nature of the solutions (real, complex, or repeated).
• Investigate how to graph quadratic equations and how the roots correspond to the points where the graph intersects the x-axis.
• Learn about applications of quadratic equations in various fields like physics, engineering, and economics.