#Solve complex quadratic equation how to

īut for the square root of a negative number, Euler (1707 – 1783) introduced the symbol i (iota) for the positive square root of – 1, i.e., i = −1. Naturally, the square of a real number is always non-negative, e.g. The concept of complex numbers came into the picture because of a few doubts, such as what if the solution of an equation does not lie on the number line? For example, what is the square root of a negative number? And so on. Additionally, complex numbers are used to find the square root of negative numbers. Hence, a complex number is the sum of a real and imaginary number. Here, a and b are real numbers, and i is the imaginary constant iota, such that i 2 = -1. What are Complex Numbers?Ī complex number can be represented in the form of a+ib. The quadratic equation is then given by x 2 – S x + P = 0. If α,β are the roots of the quadratic equation, then the sum of the roots (α+β)= -b/a and the product of the roots (α.β)= c/aĪdditionally, P and S are the product of roots and sum of roots, respectively, of a quadratic equation. Hence to know Quadratic equations in a complex number system – the discriminant must be NEGATIVE. Thus it is quite obvious that the total roots are two in each case. Negative discriminant: 𝑏 2 −4𝑎𝑐<0, no real roots or, as we say, two distinct complex roots Zero discriminant: 𝑏 2 −4𝑎𝑐=0, one repeated real root

Positive discriminant: 𝑏 2 −4𝑎𝑐>0, two real roots Using the discriminant, we identify the three different cases of quadratic equations as follows( note that we are considering only real a, b and c):

Depending on its sign, the equation may have real, complex, or coincident roots. The term b 2 – 4ac is referred to as the discriminant of the equation. Quadratic equations in a complex number systemįor a quadratic equation 𝑎𝑥 2 +𝑏𝑥+𝑐=0 with 𝑎≠0, the solutions/roots are given by 𝑥=(−𝑏±√(𝑏 2 −4𝑎𝑐))/a. Not only a but b and c can also be complex, even if they are real, the equation may have complex roots, in which case the roots are conjugate pairs. The first condition for an equation to be a quadratic equation is that the coefficient of x 2 is non-zero i.e. Thus, a quadratic equation that is a polynomial equation of order 2 has two solutions.Īnother, more used form of the quadratic equation is ax² + bx + c = 0.Ī,b and c are the constant coefficients, x is the variable. Any equation that is written in a standard form of a 1 x (n-1) + a 2 x (n-2) + a 3 x (n-3) + … + a n = 0 is a polynomial equation of degree ‘n’, thus, it has ‘n’ roots. It must be noted that the roots can be real, complex as well as there can be repeated roots.

$solve complex quadratic equation$

“A polynomial equation of degree n has exactly n roots” Fundamental Theorem of Algebra states that What is a Quadratic Equation?Ī quadratic equation is a mathematical expression of degree 2.

#Solve complex quadratic equation how to

At the same time, we will discuss quadratic equations and complex numbers individually before explaining how to solve quadratic equations in a complex number system. In this article, we will cover the generality of quadratic equations, quadratic equations with complex numbers in detail. That means, the solution of such equations includes complex numbers. Some roots might consist of imaginary numbers and these are referred to as complex roots. Introductionĭepending on the coefficients, a quadratic equation may have none, one or two real roots. Learn more about the basics of quadratic equations with complex numbers and easy tips and tricks to solve Quadratic equations examples. It will always have 2 roots/zeros/solutions. An equation of the form a 0 x 0 + a 1 x 1 + a 2 x 2 = 0 is a quadratic equation. When n is 2, this expression is quadratic. Here, n is the degree of this polynomial. A polynomial expression is an expression of the form of a 0 x 0 + a 1 x 1 + …….