91Ó°ÊÓ

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\) where \(a, b,\) and \(c\) are constants and \(a eq 0\). They are called 'quadratic' because 'quadra' means 'square' in Latin, indicating the highest degree of the equation.
Quadratic equations can appear in many forms and solving them is a fundamental skill in algebra. The equation in our exercise, \(t^2 - 4t = -1\), needs to be manipulated carefully to find the value of \(t\).
One common method to solve such equations is 'completing the square', which turns the quadratic equation into a perfect square trinomial, making it easier to solve.

Algebraic manipulation encompasses the techniques we use to simplify and solve equations. Completing the square is a method involving several algebraic steps.
Let's start from the beginning!
First, we moved the constant term to the right side of the equation. This step ensures that the equation is ready for completing the square: \[t^2 - 4t = -1\]Adding 1 to both sides, we get:\[t^2 - 4t + 1 = 0\]This expression makes it easier to isolate our variable term and focus only on the quadratic and linear terms.Next, we need to create a perfect square trinomial. We identify the coefficient of the linear term, which is -4 in this case. Then, we take half of it and square it,
\[\text{Half of } -4 \text{ is } -2 \] \(\text{Squaring } -2 \text{ gives us } 4\) Finally, adjust both sides of the equation, transforming it to: \[t^2 - 4t + 4 = 3 \] Notice the '4' added to the equation. This step completes the square and is crucial for solving.

After completing the square, you obtain an equation that is easier to solve. This resulting perfect square trinomial can usually be rewritten in a simpler form.
From our example, we have transformed the equation to: \(t^2 - 4t + 4 = 3\) This expression can be factored as: \((t - 2)^2 = 3\) At this point, we will use the square root property to isolate our variable. Taking the square root of both sides, we get two possible equations: \(t - 2 = \sqrt{3}\) and \(t - 2 = -\sqrt{3}\) We solve for each case: \(t = 2 + \sqrt{3}\) or \(t = 2 - \sqrt{3}\) Through these steps, we have effectively solved for \(t\). Thus, our solutions to the original quadratic equation \(t^2 - 4t = -1\) are:- \(t = 2 + \sqrt{3}\)- \(t = 2 - \sqrt{3}\) Following these steps ensures accurate solutions and a deeper understanding of the mathematical principles involved.