Solving Quadratic Equations: Techniques and Applications

Quadratic equations are a fundamental part of algebra and mathematics, playing a crucial role in various fields such as physics, engineering, and economics. One common quadratic equation is x2 – 2x – 15 0. Let's explore different methods to solve this equation and other similar problems.

Solving the Quadratic Equation: x2 – 2x – 15 0

Let's solve the equation x2 – 2x – 15 0 using multiple methods:

Factoring Method

The first approach involves factoring the quadratic equation. We start with the equation:

x2 – 2x – 15 0

Factor x2 – 5x 3x – 15 0.

x(x – 5) 3(x – 5) 0

(x 3)(x – 5) 0

Setting each factor to zero gives us the roots: x -5 and x 3.

Completing the Square

An alternative method to solve the same equation is by completing the square:

x2 – 2x – 15 0

Move the constant to the right side:

x2 - 2x 15

Add the square of half the coefficient of x to both sides:

x2 - 2x 1 15 1

(x - 1)2 16

Take the square root of both sides:

x - 1 ±√16

x 1 ± 4

The roots are: x 5 and x -3.

Quadratic Formula Method

The Quadratic Formula is a reliable way to solve any quadratic equation of the form ax2 bx c 0. For x2 – 2x – 15 0, the coefficients are a 1, b -2, and c -15. The formula is:

x [-b ± √(b2 – 4ac)] / (2a)

Substituting the values:

x [2 ± √((-2)2 – 4 × 1 × -15)] / (2 × 1)

x [2 ± √(4 60)] / 2

x [2 ± √64] / 2

x [2 ± 8] / 2

x 5 and x -3

Practical Applications of Quadratic Equations

Quadratic equations have numerous practical applications:

Physics: Calculating trajectories, motion under gravity.

Engineering: Designing structures, optimizing surfaces, and solving fluid dynamics problems.

Economics: Pricing models, supply and demand analysis, and optimizing costs.

Architecture: Creating parabolic arches and designing symmetric designs.

Conclusion

Solving quadratic equations is a fundamental skill in mathematics. The methods discussed here provide different approaches to solving such equations, allowing flexibility and adaptability in problem-solving situations. Whether you are factoring, completing the square, or using the quadratic formula, the goal is to find the roots of the equation efficiently.

Keywords: quadratic equation, completing the square, quadratic formula