Mastering Quadratic Equations | Understanding, Solving, and Real-World Applications

Quadratic

In mathematics, a quadratic is a polynomial equation of degree 2

In mathematics, a quadratic is a polynomial equation of degree 2. It is characterized by having a term with a variable raised to the power of 2 (x^2), and can also contain terms with a variable raised to the power of 1 (x) and a constant term (c). The general form of a quadratic equation is:

ax^2 + bx + c = 0

Here, a, b, and c are coefficients, with a ≠ 0. The variable x represents the unknown, and the goal is often to solve for its values that satisfy the equation.

The solutions to a quadratic equation can be found through various methods, including factoring, completing the square, or using the quadratic formula. Factoring involves finding two binomial expressions that, when multiplied, give the quadratic equation. Completing the square involves manipulating the equation to express it in a perfect square trinomial form. The quadratic formula is a formula that directly gives the solutions of a quadratic equation:

x = (-b ± √(b^2 – 4ac)) / (2a)

The term inside the square root, b^2 – 4ac, is called the discriminant. It determines the nature of the solutions of the quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If it is zero, the equation has a single real solution (known as a double root). And if the discriminant is negative, the equation has two complex conjugate solutions (involving the imaginary unit i).

Quadratic equations have broad applications in various fields such as physics, engineering, computer science, and economics. They are used to model a wide range of phenomena, including projectile motion, optimization problems, and financial analysis.

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