91Ó°ÊÓ

In solving quadratic equations, encountering complex solutions often happens when the discriminant is negative. But what are complex solutions? They are numbers that have a real part and an imaginary part. Imaginary numbers use the unit \(i\), which is defined as the square root of -1. When you get a solution with a square root of a negative number, like \(-23\), this indicates the presence of complex numbers in the solutions.

• The real part in our example is \( \frac{3}{8} \)
• The imaginary part is \( \frac{i\sqrt{23}}{8} \)

Thus, the solutions are written as \( z = \frac{3}{8} \pm \frac{i\sqrt{23}}{8} \). This means there are two complex solutions for the quadratic equation, differing by the +/- sign. Understanding this concept helps in grasping the nature of roots that are not real.

The quadratic formula is a key component in solving quadratic equations. It is a reliable method to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]To use the quadratic formula effectively, it’s essential to correctly identify the coefficients \(a\), \(b\), and \(c\) from the equation:

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

For the equation \( 4z^2 - 3z + 2 = 0 \), the coefficients are \(a = 4\), \(b = -3\), and \(c = 2\). You substitute these values into the formula to calculate the roots. This formula is versatile because it works for all types of solutions: real, repeated, or complex.

The discriminant is a part of the quadratic formula under the square root sign, represented as \( D = b^2 - 4ac \). It tells us about the nature of the roots or solutions of the quadratic equation.

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is exactly one real solution, often called a repeated root.
• If \(D < 0\), as in the current exercise \((-23)\), the solutions are complex.

So, when you calculate the discriminant and find that it is negative, you should expect the solutions to be in the form of complex numbers. Calculating the discriminant first can help you set expectations about the type of solutions you will get, which is essential in solving quadratics effectively.