91Ó°ÊÓ

To begin solving a quadratic equation, the first step is to isolate the variable term. In our example, the equation given is: $$ 3t^2 - 1 = 6 $$ The variable term here is \( 3t^2 \) and we want to isolate it. To do this, we need to move the constant term on the left side of the equation to the right side. We do this by adding 1 to both sides to cancel out the -1 on the left. This operation preserves the equality as long as we perform it on both sides of the equation. Consider: \( 3t^2 - 1 + 1 = 6 + 1 \) After simplifying, we get: \( 3t^2 = 7 \) Now the variable term \( t^2 \) is isolated.

Once we have isolated the variable term, the next step is to solve for the variable. In quadratic equations, this often involves taking the square root. From our simplified equation: \( t^2 = \frac{7}{3} \) We take the square root of both sides to find the values of \( t \): \( t = \text{±} \text{√} \frac{7}{3} \) The ± symbol indicates that there are two possible values—one positive and one negative. This step is crucial because squaring is an operation that affects both the positive and negative values of a number. For example:

• The square root of 4 is both 2 and -2 because \( 2^2 = 4 \) and \( (-2)^2 = 4 \).

Similarly, for our equation, we have: \( t = \frac{\text{±} \text{√} 7}{\text{√} 3} = \text{±} \text{√} \frac{7}{3} \)

The final step often involves simplifying the equation to make it more interpretable. Simplification means cleaning up the expression, combining like terms, or reducing fractions. In our case: \( t = \text{±} \text{√} \frac{7}{3} \) This represents the simplest form of our answer. We can leave it as it is or, if we prefer, rationalize the denominator for a cleaner look: \( t = \frac{\text{±} \text{√ 7}}{\text{√ 3}} \times \frac{\text{√ 3}}{\text{√ 3}} = \frac{\text{±} \text{√ 21}}{3} \) Both forms are correct, but sometimes rationalizing the denominator is more conventional in mathematical contexts. Simplifying correctly helps make the solution clearer and easier to understand. Recall:

• Simplification includes operations like dividing, factoring, and rationalizing.
• It ensures the equation is in its simplest and most interpretable form.