Nature of the Roots of the Quadratic Equation

In mathematics, a quadratic equation is any polynomial equation involving only degree 2, with no linear or higher-order terms. These polynomial equations have the form of \(a{x^2} + bx + c = 0,\) where \((a≠0)\). The roots of the quadratic equation are an algebraic expression that defines the x-intercepts of a quadratic function. The nature of the roots of the quadratic equation is that when these expressions are plotted on a graph, they intersect at two distinct points that defines the x-intercepts of a quadratic function.

The roots of a quadratic equation are the solutions that satisfy the equation. In other words, if we have a quadratic equation with real solutions, then the roots of this equation will be real numbers. However, the way to find the roots of a quadratic equation is by factoring it into two other equations with one unknown and one constant. Then we solve these two equations to find the value of our unknown roots.

What is a Quadratic Equation?

A polynomial equation of the second degree in x is called a quadratic equation. Such equations have the standard form \(a{x^2} + bx + c = 0\), where a and b are coefficients, \(x\) is the variable, and \(c\) is the constant term. The coefficient of \(x^2\) is a non-zero term \((a≠0)\) and is the first measure for determining whether an equation is quadratic.

Quadratic Equation Formula

Let us take a look at how to find the roots (α, β) of the quadratic equation. First, we need to look at the general formula for solving quadratic equations. The alpha (α) and beta (β) symbols stand for representing the roots of a quadratic equation. The following quadratic equation formula helps in solving and finding the roots of those quadratic equations which are difficult to factorise:

\(x = \frac{{ – b \pm \sqrt {{b^2} – 4ac} }}{{2a}}\)

The expression under the square root in the above formula is known as Discriminant. The Discriminant, represented by \(D or Δ\) in the quadratic formula. It helps determine the nature of the roots of the quadratic roots.

what is discriminant in a quadratic equation

Nature of the Roots of the Quadratic Equation

The value of the Discriminant, \(D=({b^2} – 4ac)\), determines the nature of the roots of the quadratic equation. If \(a, b, c ∈ R\), where \(R\) belongs to real numbers, then the roots of the second-degree equation can either be real or imaginary according to the following criteria:

Two distinct real roots, if \({b^2} – 4ac > 0\)
Two equal real roots, if \({b^2} – 4ac = 0\)
No real roots if \({b^2} – 4ac < 0\)

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