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When you encounter a quadratic equation, it typically means that you are dealing with expressions where the variable is squared, like \( ax^2 + bx + c = 0 \). These equations can often be solved by factoring, using the quadratic formula, or completing the square. In our example, we are using the factoring method, which involves breaking down the equation into simpler parts. When you have an equation structured like \((t + 8)(t - 6) = 0\), this is a classic case for the zero-product property. You tackle it by solving for the variable in each factor individually. Here, we solve for \( t \) in both \( t + 8 = 0 \) and \( t - 6 = 0 \). By solving these linear equations, we can find the values of \( t \) that make the product zero.

Factoring plays a crucial role in solving quadratic equations. Through factoring, it becomes possible to express the equation as the product of its linear components, as seen in \((t + 8)(t - 6) = 0\).

• Identifying Factors: Look for numbers or expressions that multiply to reach the original equation. Here, notice \((t + 8)\) and \((t - 6)\) are chosen such that their product equals zero.
• Zero-Product Property: This property states that if the product of two expressions is zero, then one or both of the expressions must be zero. Therefore, you solve by setting each factor equal to zero.

By doing this, you uncover potential solutions to the initial equation, such as \( t = -8 \) and \( t = 6 \), which deliver the conditions where the product could be nullified.

After solving for the potential solutions of a quadratic equation, it is essential to verify them. Verification ensures that the obtained solutions satisfy the original equation. Here's how verification works:

• Substitute each solution back into the original equation.
• Check if the equation holds true for these values.

For instance, when substituting \( t = -8 \) in our equation \((t+8)(t-6)\), it results in \((0)(-14) = 0\). Similarly, for \( t = 6 \), it results in \((14)(0) = 0\). Final check: if both cases give zero, the solutions are verified as correct!Verification is not just a procedural step; it's a vital part of solving any equation, confirming that the problem-solving process was thoroughly accurate.