Quadratic equations are mathematical expressions that involve a variable raised to the second power (x²) and may also include a linear (x) and constant term. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is the variable. These equations can have one, two, or no real solutions, depending on the values of the constants and the nature of the roots (real or complex).
They are widely used in various fields of mathematics and science to model real-life situations, such as projectile motion, optimization problems, and electrical engineering applications. Quadratic equations can be solved using different methods, including factoring, completing the square, and using the quadratic formula, which provides a direct way to find the roots of the equation.
Quadratic Equations and Its Roots
Roots can be defined as the values of the variables that satisfy the requirements of a given equation. x = a is said to be the roots of the quadratic equation f(x), if f(a) = 0.
The real roots of the equation f(x) = 0 can be said so because of the x-coordinates of the points; the point where the curve y = f(x) intersects the x-axis.
It's proved that one of the roots of the quadratic equation is zero whereas the opposite is equal to -b/a if c = 0.
In case, b = c = 0 then, each of the roots is measured to be zero.
Once a = c, the roots are reciprocal to one another.
Nature of the Roots of Quadratic Equations
If the value of the discriminant is equal to 0 i.e. b2 – 4ac = 0
In this case, the quadratic equation will have equal roots i.e. a = b = -b / 2a
If the value of discriminant < 0 xss=removed xss=removed> 0 i.e. b2 – 4ac > 0
In this case, the quadratic equations will have real roots.
If the worth of the discriminant > 0 and D is found to be a perfect square.
In this case, the quadratic equation can have rational roots.
If the value of the discriminant is greater than 0 and D isn't a perfect square.
In this case, the quadratic equation can have irrational roots i.e. a = (p + √q) and b = (p – √q)
If the value of the discriminant is greater than 0 and D is found to be a perfect square.
Here, a = 1 and b & c can be termed as integers.
In this case, the quadratic equation goes to have integral roots.