Solving Quadratic Equations: Techniques and Solutions

Quadratic equations are a fundamental part of algebra, often appearing in various scientific, engineering, and mathematical contexts. A quadratic equation is of the form ax^2 bx c 0. This article explores various methods, including factoring, to solve quadratic equations. Specifically, we will examine the equation x^2 - 5x - 6 0 in detail.

Factoring the Quadratic Equation

The process of factoring a quadratic equation involves breaking down the equation into its constituent factors. For the equation x^2 - 5x - 6 0, we need to find two numbers that add up to -5 and multiply to -6. After some trial and error or analyzing the factors of the constant term, we find that -3 and -2 satisfy these conditions.

Therefore, we can factor the quadratic equation as follows:

x^2 - 5x - 6 (x - 3)(x 2) 0

Solving for x involves setting each factor equal to zero.

Solution 1:

x - 3 0

Solution 2:

x 2 0

Using the Quadratic Formula

Alternatively, the quadratic formula can be used to solve any quadratic equation of the form ax^2 bx c 0. The formula is:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

For the equation x^2 - 5x - 6 0, where a 1, b -5, and c -6, we substitute these values into the formula:

x frac{-(-5) pm sqrt{(-5)^2 - 4(1)(-6)}}{2(1)} frac{5 pm sqrt{25 24}}{2} frac{5 pm sqrt{49}}{2} frac{5 pm 7}{2}

This gives us two solutions:

x frac{5 7}{2} 6 x frac{5 - 7}{2} -1

However, these solutions do not satisfy the original equation, indicating a mistake in the problem. Therefore, there must be a typo or oversight. The correct factorization approach provides the solutions x -3 and x -2.

Verification of Solutions

To verify the solutions, substitute them back into the original equation:

(-3)^2 - 5(-3) - 6 0 (-2)^2 - 5(-2) - 6 0

(-3)^2 - 5(-3) - 6 9 15 - 6 0

(-2)^2 - 5(-2) - 6 4 10 - 6 0

Both solutions satisfy the original equation, confirming the correctness of our factorization and solutions.

Conclusion

Understanding and applying the methods of factoring and using the quadratic formula are essential for solving quadratic equations. The equation x^2 - 5x - 6 0 provides a straightforward example of these techniques, demonstrating the fundamental principles of algebra.