Solve 2x^2 - 11x - 14 0: Methods and Techniques for Finding Roots

Introduction

The quadratic equation 2x^2 - 11x - 14 0 is a common algebraic problem that can be solved using various methods. This article will explore two primary techniques: the quadratic formula and factoring, with detailed steps and explanations for each method.

Solving the Quadratic Equation Using the Quadratic Formula

The quadratic formula, a powerful tool for solving quadratic equations, is given by:

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

Step-by-Step Solution

Identify the coefficients: a 2 b -11 c -14 Calculate the discriminant: b^2 - 4ac (-11)^2 - 4(2)(-14) 121 112 233 Apply the quadratic formula: x frac{-(-11) pm sqrt{233}}{2(2)} frac{11 pm sqrt{233}}{4} Find the two solutions: x_1 frac{11 sqrt{233}}{4} x_2 frac{11 - sqrt{233}}{4}

Solving the Quadratic Equation by Factoring

Factoring is another method that can be used to solve quadratic equations. In this method, we look for two numbers that multiply to ac 2(-14) -28 and add to b -11.

Find the correct pair of factors: -14 and 2 (since -14 * 2 -28 and -14 2 -12) Rewrite the quadratic expression: 2x^2 - 14x 2x - 14 Group the terms: (2x^2 - 14x) (2x - 14) Factor by grouping: 2x(x - 7) 2(x - 7) (2x 2)(x - 7) Solve for x by setting each factor to zero: 2x 2 0 implies x -1 x - 7 0 implies x 7

Summary

Both methods yield the same solutions:

x frac{11 sqrt{233}}{4} x frac{11 - sqrt{233}}{4}

Conclusion

Solving quadratic equations is a fundamental skill in algebra. By using methods like the quadratic formula and factoring, you can efficiently find the roots of any quadratic equation. Understanding these techniques not only enhances your problem-solving abilities but also provides a solid foundation for more advanced mathematical concepts.