(a) Compute the discriminant of the quadratic and note that...

(a) Compute the discriminant of the quadratic and note that it is negative (and therefore the equation has no real-number roots).
(b) Use the quadratic formula to obtain the two complex conjugate roots of each equation.

$$-10 z^{2}+4 z-2=0$$

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Key Concepts

Complex Conjugate Roots

When the discriminant of a quadratic equation is negative, the square root of a negative number results in an imaginary number. This leads to the formation of complex conjugate roots, which are pairs in the form a + bi and a - bi, ensuring that any real-coefficient polynomial equation retains real parts in its solution.

Quadratic Equation

A quadratic equation is a second-degree polynomial expressed in the form ax² + bx + c = 0, where a is nonzero. It is a fundamental concept in algebra because it represents parabolic curves and is used to model various scenarios in mathematics and science.

Discriminant

The discriminant, given by the formula b² - 4ac in a quadratic equation, determines the nature of its roots. A positive discriminant indicates two distinct real roots, zero implies a double real root, and a negative discriminant reveals that the roots are complex, occurring as conjugate pairs.

Quadratic Formula

The quadratic formula, x = (-b ± ?(b² - 4ac)) / (2a), is a universal method for solving quadratic equations. It utilizes the coefficients of the equation along with the discriminant to provide the exact solutions, whether they are real or complex.

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