91Ó°ÊÓ

The square root property is a useful tool in solving quadratic equations. It states that if we have an equation of the form \(a^2 = b\), we can take the square root of both sides to find \(a = \pm \sqrt{b}\). This property helps us eliminate the squared term and simplify the equation.
In the given exercise, we apply the square root property to the equation \((5t + 3)^2 = 36\). By taking the square root of both sides, we get two solutions: positive and negative. This results from \(\sqrt{(5t + 3)^2} = \pm \sqrt{36}\), leading to \(5t + 3 = \pm 6\). This step reduces our quadratic equation into two linear equations that we can solve easily.

Simplifying radicals is essential when solving equations involving square roots. A radical is in its simplified form when the radicand (the number under the radical sign) has no perfect square factors, other than 1.
In this exercise, we encountered the term \(\sqrt{36}\). To simplify, we recognize that 36 is a perfect square, and its square root is 6. Thus, \(\sqrt{36} = 6\). Simplifying radicals ensures that our solutions are in their simplest forms, making them easier to understand and work with.

When solving quadratic equations, concepts like the square root property and simplifying radicals often come into play. Quadratic equations typically have two solutions because they are second-degree polynomials. In the example problem, after applying the square root property, we ended up with two linear equations: \(5t + 3 = 6\) and \(5t + 3 = -6\).
For \(5t + 3 = 6\), we solve for \(t\) by isolating the variable. Subtract 3 from both sides to get \(5t = 3\), and then divide by 5, resulting in \(t = \frac{3}{5}\).
For \(5t + 3 = -6\), we follow a similar process. Subtract 3 from both sides to get \(5t = -9\), and then divide by 5, leading to \(t = -\frac{9}{5}\).
These steps allow us to find the two solutions to the quadratic equation, ensuring that we've fully solved the problem and understood the dynamics involved in quadratic solutions.