91Ó°ÊÓ

The quadratic formula is an essential tool for solving quadratic equations, especially when they do not factor easily. A quadratic equation is generally expressed in the form \(ax^2 + bx + c = 0\), where "a", "b", and "c" are constants. The quadratic formula is given by:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula allows you to find the roots of any quadratic equation. Simply substitute the values for "a", "b", and "c" into the formula, and solve for "x".

The expression inside the square root, \(b^2 - 4ac\), is known as the discriminant. The discriminant determines the nature of the roots. If the discriminant is positive, there are two distinct real solutions. When it is zero, there is exactly one real solution. A negative discriminant means the solutions are complex, involving imaginary numbers.

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). This is the most common form used in quadratic equations because it allows for straightforward application of solution techniques, such as factoring, completing the square, or using the quadratic formula.

In our example, we started with the equation \(5x^2 = 2x - 3\). To convert it into standard form, we need all terms on one side of the equation, resulting in \(5x^2 - 2x + 3 = 0\). This rearrangement sets up the equation for further analysis, emphasizing the role of each coefficient in solving the equation.

Rewriting in standard form is crucial as it not only simplifies the process of finding solutions but also helps in identifying key values ("a", "b", and "c") for the quadratic formula, and showcases the balance of the equation.

When the discriminant of a quadratic equation (\(b^2 - 4ac\)) is negative, the quadratic equation has complex solutions. Unlike real solutions, complex solutions involve imaginary numbers, which are based on the imaginary unit \(i\), where \(i^2 = -1\).

In the exercise given, the value of the discriminant is \((-2)^2 - 4 \times 5 \times 3 = 4 - 60 = -56\). Since \(-56\) is less than zero, the solutions involve the square root of a negative number, resulting in complex numbers:

• \(x = \frac{2 \pm \sqrt{-56}}{10}\)
• Simplifies to \(x = \frac{2 \pm \sqrt{56}i}{10}\)

These solutions, \(x = \frac{2 \, + \, \sqrt{56}i}{10}\) and \(x = \frac{2 \, - \, \sqrt{56}i}{10}\), are conjugates, demonstrating that complex solutions always come in pairs. Understanding complex solutions is vital for complex number theory, extending the realm of mathematics beyond just real numbers.