91Ó°ÊÓ

The quadratic formula is a powerful tool for solving any quadratic equation, which generally takes the form \(ax^2 + bx + c = 0\). It allows us to find the roots, or solutions, of the equation without needing to factor. The formula is:

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

Here:

• \(a\), \(b\), and \(c\) are coefficients from the equation.
• The symbol \(\pm\) indicates there will be two solutions: one using addition, and the other, subtraction.

The key is to carefully substitute these values and simplify, which opens the door to unraveling the mystery of quadratic equations.
Once you’ve plugged in your values, the quadratic formula can lead you to two potential types of solutions: real or complex. It all depends on the term under the square root, known as the discriminant \((b^2 - 4ac)\).
In this exercise, after plugging in \(a = 1\), \(b = -16\), and \(c = 65\), the equation becomes \(t = \frac{16 \pm \sqrt{-4}}{2}\), leading to complex solutions due to the negative discriminant.

When we encounter a negative number under a square root, we enter the realm of imaginary numbers, which are necessary for expressions like \(\sqrt{-4}\).
An imaginary number is the square root of a negative number, symbolized by 'i', where \(i = \sqrt{-1}\). It's a concept that is especially useful in complex solutions.
In this equation, \(\sqrt{-4}\) can be expanded to \(\sqrt{4} \cdot \sqrt{-1} = 2i\). Here, \(\sqrt{-1}\) is represented by \(i\).
Imaginary numbers are not "imaginary" in the sense of being fictional; they allow mathematics to deal with real-world problems where complex numbers naturally arise, such as in engineering and physics.

• i is the fundamental imaginary unit.
• It helps in solving equations that don't have real solutions.
• Combines with real numbers to form complex numbers.

Complex solutions arise when the solutions to a quadratic equation involve imaginary numbers. A complex number has two parts: a real part and an imaginary part.
For example, using the previous calculation, the roots for the equation \(t^2 - 16t + 65 = 0\) are \(8 + i\) and \(8 - i\), which are complex solutions.
Each solution here comprises:

• A real part: 8.
• An imaginary part: \(\pm i\).

These roots can still be represented on a complex plane, which is like a coordinate plane for complex numbers. On this plane:

• The horizontal axis represents the real part.
• The vertical axis represents the imaginary part.

Complex solutions reveal the rich potential of mathematics to provide solutions even when those solutions don’t lie on the number line of real numbers. They extend our understanding and ability to solve quadratic equations fully.