91Ó°ÊÓ

A quadratic equation is an important concept in mathematics, often expressed in the standard form of \( ax^2 + bx + c = 0 \). These equations represent parabolas when graphed on a coordinate plane.
Understanding the different parts of a quadratic equation is crucial. Here, the term \( ax^2 \) is what defines it as 'quadratic'—the term 'quadratic' comes from 'quad,' meaning square. Each letter in the equation has a role:

• \( a \) is the coefficient of \( x^2 \) and affects the parabola's width and direction (upwards if positive, downwards if negative).
• \( b \) is the coefficient of \( x \) and affects the direction and position of the parabola along the x-axis.
• \( c \) represents the y-intercept where the parabola crosses the y-axis.

In solving a quadratic equation, like the one in our original problem \( x^2 + 4x - 5 = 0 \), we set the equation equal to zero. This allows us to find the roots or solutions of the equation, which represent the x-values where the graph of the equation crosses the x-axis. These solutions can be found using several methods, including factoring, completing the square, or using the quadratic formula.

A one-to-one function is a function where each output value is paired with exactly one input value. Understanding whether a function is one-to-one is key because it guarantees the existence of an inverse function over a specific domain.
For example, in our exercise, the function \( g(x) = x^2 + 4x \) with the restriction \( x \geq -2 \) is considered one-to-one. This is due to its domain constraint, which ensures that it passes the horizontal line test. When a function restricted in this way, each y-value in the range corresponds to only one x-value in the domain, making the inverse function possible.
To check if a function is one-to-one, you can apply the horizontal line test: if no horizontal line intersects the graph of the function more than once, then the function is one-to-one.
The importance of a one-to-one function is underscored when trying to find an inverse. Since the function has an inverse, we can solve for \( g^{-1}(5) \) as we did in the exercise, ensuring that the solution \( x = 1 \) is valid and unique.

The quadratic formula is an essential tool in algebra for finding the solutions of a quadratic equation. When you cannot factor a quadratic equation easily, the quadratic formula provides a reliable way to find the roots:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, the symbols have specific meanings:

• \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).
• \( \pm \) indicates that there can be two possible solutions.
• The term \( \sqrt{b^2 - 4ac} \) is called the "discriminant," and it determines the number and type of solutions:
• If the discriminant is positive, there are two different real solutions.
• If it is zero, there is one real solution (a repeated root).
• If it is negative, there are no real solutions (the solutions are complex).

In our exercise, we used the quadratic formula to solve \( x^2 + 4x - 5 = 0 \). Plugging the coefficients into the formula, we found two potential values: \( x = 1 \) and \( x = -5 \). Since we have a domain restriction (\( x \geq -2 \)), only \( x = 1 \) is a valid solution.