![]() ![]() Of each equation plotted above the complex plane. The following figures represent the graph of the modulus Vertical or horizontal line in the complex plane. Notice that, unlike the first two examples, the two roots in the previous examples do not lie on a □ = − 4 − 3 √ 2 2 1 3 + 6 − 2 √ 2 2 1 3 □ are the solutions of the equation. Multiplying both the numerator and denominator by the complex conjugate of the denominator gives ( 2 + 3 □ ) ( 4 − 6 □ ) is twice the complex number Ĭanceling the factor of 2 in the numerator and denominator, we have Taking the 4 out of the square root, we have The fact that we have some nonreal coefficients, the discriminant is still a real number.Įxample 3: Nonreal Quadratic Equations with Real Discriminants In the first couple of examples, we will consider the special case where, in spite of The square root of the discriminant for nonreal numbers. Hence, even for quadratics with complex coefficients, we can use the quadratic formula,Īlthough we might need to appeal to de Moirve’s theorem to calculate the value of ![]() Roots and consider both the positive and negative values, which gives Hence, returning to equation (1), we can take square We can find a root of a complex number using de Moirve’s theorem, and then the second root Therefore, in the same way that we take both the positive and negative roots for real numbers, However, courtesy of the Euler’s identity, we know □ = − 1 therefore, Using the properties of exponential functions, we can express this as Using de Moivre’s theorem, the two possible square roots of this number are However, let us consider the case of a general complex number □ = □ □ . Root, is something we might need to be careful with, since taking roots of a complex number returns multiple ![]() Īll of the steps we have taken up to this point are equally valid whether □, Let us consider how to solve a general quadratic equation In this explainer, we will relax the condition that the quadratic equation has real coefficientsĪnd explore what we can conclude about the roots of quadratic equations with complex coefficients. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |