91Ó°ÊÓ

In mathematics, a quadratic equation is any equation that can be rearranged in standard form as \textrm{\textbf{ax\(^{2}\) + bx + c = 0}}, where x represents an unknown variable, and a, b, and c represent known numbers, with a ≠ 0. This equation is called 'quadratic' because the term 'quad' refers to a square, indicating that the variable is raised to the power of 2.

To solve a quadratic equation, you can use the quadratic formula: \textrm{\textbf{x = \frac{-b \text{±} \text{\textsqrt{b\(^{2}\)-4ac}}}{2a}}}. Learn the formula by heart, as it's a powerful tool for finding the roots, or solutions, of quadratic equations.

Let's break it down:

• Identify a, b, and c in the equation.
• Plug these values into the quadratic formula.
• Calculate the value inside the square root (the discriminant).
• Evaluate both the positive and negative solutions.

In our example, after substituting \textrm{\textbf{\textln x}} with y, we transformed the logarithmic equation into the quadratic equation 2y\(^{2}\) + y - 1 = 0. By identifying a=2, b=1, and c=-1, we successfully used the quadratic formula to find two solutions for y.

The substitution method is a useful problem-solving technique that makes complex equations simpler by converting them into more manageable forms. Essentially, substitution involves replacing one part of the equation with a temporary variable.

Steps for the substitution method:

• Identify a suitable substitution. For instance, \textrm{\textbf{y = \text{\textln x}}}.
• Replace the identified part of the equation with the new variable.
• Solve the transformed (often simpler) equation.
• Substitute back the original variable into your solution.

In our problem, we used substitution to ease the solution of the logarithmic equation \textrm{\textbf{2(\text{\textln x})\(^{2}\) + \text{\textln x} - 1 = 0}}. By letting y = \text{\textln x}, the logarithmic equation transformed into a quadratic equation in terms of y. Once we solved this simpler quadratic equation, we reversed our substitution, converting our solutions for y back into solutions for \text{\textln x}, and finally solved for x.

The natural logarithm, denoted as \textrm{\textbf{\text{\textln(x)}}}, is a specific logarithm having the base e, where e \text{\textapprox 2.71828} is an irrational and transcendental constant. Essentially, the natural logarithm of a number x, \text{\textln(x)}, can be defined as the power to which e must be raised to obtain that number x.

Properties of natural logarithms:

• The natural logarithm is the inverse function of the exponential function, meaning \textrm{\text{\textln(e\(^{x}\))} = x.
• For x > 0, \textrm{\text{\textln(ab)} = \text{\textln(a)} + \text{\textln(b)}}.
• The natural logarithm of 1 is 0, i.e., \textrm{\text{\textln(1)} = 0.

In our original equation, we manipulated \textrm{\textbf{\text{\textln(x)}}} directly and later exponentiated both sides of our solution to find x itself. For example, when solving \textrm{\text{\textln x} = 0.5}, we found \textrm{\textbf{x = e\(^{0.5}\) = \text{\textsqrt{e}}}}.
Likewise, for \textrm{\text{\textln x} = -1}, the solution was \textrm{\textbf{x = e\(^{-1}\) = \text{\text1/e}}}, effectively utilizing the inherently inverse relationship between natural logarithms and exponentials.