91Ó°ÊÓ

The quadratic formula is a powerful tool used to solve quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). It is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \(a\), \(b\), and \(c\) are coefficients of the equation, with \(a eq 0\). The term under the square root, \(b^2 - 4ac\), is called the "discriminant."

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is exactly one real solution.
• If it is negative, there are no real solutions.

In the given equation, after converting from logarithmic to exponential form, the quadratic equation was: \(x^2 - 8x - 9 = 0\). Using the quadratic formula yields potential solutions \(x = 9\) and \(x = -1\). It's important to substitute these back into the original problem to check for validity.

Converting a logarithmic equation into its exponential form is a crucial step for solving it algebraically. A logarithmic equation such as \(\log_b y = x\) can be rewritten as an exponential equation, \(b^x = y\).

This conversion simplifies further calculation, allowing you to operate directly in terms of powers and bases. In the case of the exercise, the equation \(\log_3 (x^2 - 8x) = 2\) was converted to exponential form: \(3^2 = x^2 - 8x\).

By rewriting it this way, we immediately see what \(x^2 - 8x\) needs to equal (in this case, 9), making it much easier to solve for \(x\). Recognizing this form helps bridge the gap between logarithmic and standard algebraic techniques.

The addition property of logarithms is a key concept that allows you to simplify expressions involving logarithms. It states that the sum of two logarithms with the same base can be combined into a single logarithm:
\[\log_b(m) + \log_b(n) = \log_b(mn)\]
This property is instrumental when you're trying to combine logarithmic expressions into a simpler form that can be worked with algebraically.

In the exercise, the initial logarithmic equation \(\log_3 x + \log_3(x - 8) = 2\) was combined using the addition property into a single logarithmic equation: \(\log_3 (x(x - 8)) = 2\).

This simplification prepared the equation for transformation using exponential form, making it easier to manipulate and solve completely.

Extraneous solutions are solutions that arise during the problem-solving process, but are not valid for the original equation. They often occur when both sides of an equation are squared (or other non-linear transformations) or when certain domain restrictions are ignored.

In this logarithmic equation exercise, squaring and simplifying processes allowed for the potential solutions \(x = 9\) and \(x = -1\).

• For \(x = -1\), when substituted back into \(\log_3(x - 8)\), results in a logarithmic function of a negative number, which is undefined. This makes \(x = -1\) an extraneous solution.
• For \(x = 9\), it satisfies \(\log_3(x - 8)\) as \(x = 9\) calculates a positive argument, thus it remains a valid solution.

Always validate solutions within the constraints of the original problem to ensure they are truly viable.