Quadratic Equations - Summary
A quadratic equation in the variable x is a mathematical expression that takes the form ax2 + bx + c = 0, where a, b, and c are real numbers, and a isn’t equal to zero. Quadratic equations are the polynomial equations of degree 2 in one variable, making them essential in math. This means they can be written as f(x) = ax2 + bx + c = 0, with a, b, c ∈ R and a ≠ 0. This is the general representation of a quadratic equation where ‘a’ is the leading coefficient, and ‘c’ is known as the absolute term of f(x). The solutions for x that satisfy this quadratic equation are referred to as the roots of the quadratic equation (α, β).
Understanding Quadratic Equations
A quadratic polynomial becomes a quadratic equation when it is set to zero. The values of x that satisfy this equation are its roots.
Quadratic Equation Formula
The solutions, or roots, of a quadratic equation can be found using the quadratic formula:
(α, β) = [-b ± √(b2 – 4ac)]/2a
Roots of Quadratic Equations
The values of the variables that satisfy the quadratic equation are called its roots. Specifically, x = α is a root of the quadratic equation f(x) if f(α) = 0.
The real roots of the equation f(x) = 0 are where the curve y = f(x) intersects the x-axis.
- One root is zero, and the other is -b/a if c = 0.
- Both roots are zero if b = c = 0.
- The roots are reciprocal to each other if a = c.
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