Quadratic Equations and Their Roots, Discriminant: A quadratic equation is a mathematical expression of the form: $$ax^2 + bx + c = 0$$ where a, b, and c ar...
Quadratic Equations and Their Roots, Discriminant:
A quadratic equation is a mathematical expression of the form:
where a, b, and c are constants.
A quadratic equation has two distinct roots, which can be found by solving for x by factoring the quadratic expression or using the quadratic formula:
The discriminant of a quadratic equation is a numerical value that helps determine the number and nature of its roots. It is calculated as:
The discriminant of a quadratic equation determines the number and nature of its roots:
If b^2 - 4ac > 0, the equation has two distinct real roots.
If b^2 - 4ac = 0, the equation has one repeated real root (also known as a double root).
If b^2 - 4ac < 0, the equation has two complex roots.
Examples:
This equation has two distinct real roots, which can be found using the quadratic formula:
This gives the roots as x = 1 and x = 3.
This equation has one repeated real root (which is also equal to 1), which can be found by factoring the quadratic expression or using the quadratic formula.
The discriminant of this equation is:
Since the discriminant is zero, the equation has two complex roots