91Ó°ÊÓ

Quadratic equations sometimes have solutions that are not real numbers. These are known as complex numbers. Complex numbers are of the form \((a + bi)\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \((i^2 = -1)\). In the given exercise, we found real solutions, but it's essential to know how to work with complex solutions too. When solving quadratics with negative numbers inside the square root, you'll encounter imaginary numbers. For example, \(\frac{-1}{6}\), which would lead to complex solutions like \(t = \pm \sqrt{-\frac{5}{6}}\) or \(t = \pm\frac{\sqrt{5}}{\sqrt{-6}}\). To make it even simpler: \(\frac{\sqrt{5}}{\sqrt{6}}\times\frac{\sqrt{6}}{\sqrt{6}}= \frac{\sqrt{30}}{-\frac{6}{36}}= \frac{+ - j}{6}\)

Understanding square roots is critical when solving quadratic equations. A square root of a number \(x\) is a value \(y\) such that \(y^2 = x\). Every positive number has two square roots: one positive and one negative. For instance, \( \sqrt{4} = 2\) and \( \sqrt{4} = -2\). In the process of solving quadratic equations, extracting square roots allows us to find both possible values for the variable. For example, when solving \( t^2 = \frac{5}{6}\), the solutions are \( t = \pm \sqrt{\frac{5}{6}}\).

Quadratic equations are usually written in standard form, which looks like this: \( ax^2 + bx + c = 0 \). This form makes it easier to apply various solution methods like factoring, completing the square, and using the quadratic formula. In the given problem, the equation \( 6t^2 - 5 = 0 \) can be rewritten in standard form as \( 6t^2 - 5 + 0 = 0 \). This standard form lets us easily understand and manipulate the equation to find the solutions.

When solving quadratic equations, it's crucial to consider both positive and negative roots. Square roots have two solutions: one positive and one negative. For example, in the equation \( t^2 = \frac{5}{6}\), when we take the square root of both sides, we get two possible solutions: \( t = \sqrt{\frac{5}{6}} \) and \( t = -\sqrt{\frac{5}{6}} \). Always remember to include both roots in your final answer to ensure you're considering all possible solutions to the equation.