(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.

$$2 z^{2}+3 z+4=0$$

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Key Concepts

Discriminant

The discriminant of a quadratic equation, given by the expression b² - 4ac, is crucial for determining the nature of the roots. A positive discriminant indicates two distinct real roots, zero indicates one repeated real root, and a negative discriminant indicates that the roots are complex numbers, as it requires taking the square root of a negative number.

Quadratic Formula

The quadratic formula is a standard method used to find the roots of any quadratic equation of the form ax² + bx + c = 0. It is expressed as (-b ± ?(b² - 4ac)) / (2a). When the discriminant (b² - 4ac) is negative, the formula still provides valid roots in the complex number system, using the imaginary unit to handle the square root of negative numbers.

Complex Conjugates

In equations with real coefficients that result in a negative discriminant, the quadratic formula yields two roots that are complex conjugates. This means that if one root is expressed as a + bi, the other will be a - bi, maintaining the symmetry and structure of solutions in the complex plane.

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