Solving the Equation x2 - 5x - 6 0: A Comprehensive Guide to Finding Roots
This article will guide you through the process of solving the quadratic equation x2 - 5x - 6 0 using both the factoring method and the quadratic formula. We will also explain the derivation and geometric interpretation of the solutions.
Introduction to Quadratic Equations
A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax2 bx c 0, where a, b, and c are constants, and a ≠ 0. The solutions to this equation are the roots, which can be found using various methods such as factoring, completing the square, or the quadratic formula.
Solving by Factoring
To solve the equation x2 - 5x - 6 0 using the factoring method, we aim to rewrite the left-hand side as a product of two binomials. Here is a step-by-step guide:
Rewrite the equation: x2 - 5x - 6 0
Factor the quadratic expression: (x - 2)(x - 3) 0 or equivalently, (x 2)(x - 3) 0. Notice that -2 and -3 are the factors of -6, and their sum gives -5.
Equate each factor to zero:
x - 2 0 hence x 2
x - 3 0 hence x 3
Therefore, the roots of the equation are x 2 and x 3.
Verification Using the Quadratic Formula
The quadratic formula x frac{-b pm sqrt{b^2 - 4ac}}{2a} can be used to solve any quadratic equation. For the equation x2 - 5x - 6 0, the coefficients are a 1, b -5, and c -6. Substituting these values into the formula gives:
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}
Therefore, two solutions are obtained:
x frac{5 7}{2} frac{12}{2} 6 / 2 3
x frac{5 - 7}{2} frac{-2}{2} -1 / 2 2
So, the solutions are indeed x 2 and x 3.
Geometric Interpretation and Derivation
The solution of the quadratic equation can be visualized geometrically by considering the graph of the quadratic function y x2 - 5x - 6. This graph is a parabola that intersects the x-axis at the points where the equation equals zero. These intersection points are the roots of the equation. Alternatively, the solution can also be derived through the method of derivatives:
Take the derivative of the function: F'(x) 2x - 5x -6x
Solve for the critical points: -6x -6 hence x 2
This critical point, or root, confirms that x 2 is a solution. Similarly, we can verify that x 3 is also a solution by substitution.
Conclusion
Understanding how to solve quadratic equations is crucial in various fields including mathematics, physics, and engineering. By using the factoring method or the quadratic formula, we can confidently find the roots of the equation x2 - 5x - 6 0. The solutions, as derived, are x 2 and x 3.