Solving the Quadratic Equation x2 - 4x - 5 0: Methods and Techniques
The quadratic equation x2 - 4x - 5 0 can be solved using multiple methods. Here, we will explore four different methods to solve this equation, catering to various levels of familiarity with algebra.
Method 1: Factoring
The equation x2 - 4x - 5 0 can be factors as follows:
Identify the coefficients: a 1, b -4, c -5. Find factors of ac (which is 1 * -5 -5) that add up to b (which is -4). The factors of -5 that add up to -4 are -5 and 1. Split the middle term using these factors:x2 - 5x x - 5 0.
Group the terms:(x2 - 5x) (x - 5) 0.
Factor out the common terms:x(x - 5) 1(x - 5) 0 or (x 1)(x - 5) 0.
Solve for x by setting each factor equal to zero:x 1 0 or x - 5 0.
Thus, x -1 or x 5.Method 2: Quadratic Formula
The quadratic formula, x -b ± √(b2 - 4ac) / 2a, can be applied to solve the equation:
Substitute the values a 1, b -4, and c -5 into the formula: x -(-4) ± √((-4)2 - 4 * 1 * -5) / 2 * 1. Simplify the expression inside the square root: x 4 ± √(16 20) / 2. Further simplify: x 4 ± √36 / 2. Finally: x 4 ± 6 / 2. This gives us two solutions for x: x 10 / 2 5 and x -2 / 2 -1.Method 3: Splitting the Polynomial
Another method involves splitting the polynomial into two sets of two terms:
Start with the equation: x2 - 4x - 5 0. Split the middle term -4x into two parts that multiply to -5 and add to -4 (e.g., -5 and 1): x2 - 5x x - 5 0. Group and factor the terms: (x2 - 5x) (x - 5) 0 x(x - 5) 1(x - 5) 0. The common factor is (x - 5): (x - 5)(x 1) 0. Set each factor equal to zero: x - 5 0 or x 1 0. Solve for x to get the solutions: x 5 or x -1.Method 4: Simplification Using Given Text
The text below provides another way to solve the quadratic equation:
Start with the equation: x2 - 4x - 5 0. Split the equation by factoring: x2 - 5x x - 5 0. Group the terms: (x2 - 5x) (x - 5) 0 Factor out the common terms: x(x - 5) 1(x - 5) 0 (x - 5)(x 1) 0. Solve for x by setting each factor equal to zero: x - 5 0 or x 1 0. Thus, x 5 or x -1.In conclusion, the solutions to the quadratic equation x2 - 4x - 5 0 are x 5 and x -1.
About the Methods:
Each method demonstrates a different approach to solving quadratic equations, emphasizing the importance of understanding various algebraic techniques. These methods can be applied to other quadratic equations with similar steps and logic.