91Ó°ÊÓ

A quadratic equation is a polynomial equation of the second degree. This means that the highest exponent of the variable, usually indicated as x, is two. The general form of a quadratic equation is written as follows:
\[ ax^2 + bx + c = 0 \]
where \(a, b,\) and \(c\) are constants, and \(a eq 0\).
In the exercise, our quadratic equation is:
\[ x^2 - 9 = 0 \]
Notice there's no \(bx\) term and \(c = -9\). This makes the equation simpler, but it's still quadratic because of the \(x^2\) term.

The standard form of a quadratic equation is exactly what we need to start solving it. A quadratic equation is in standard form if it looks like this:
\[ ax^2 + bx + c = 0 \]
For our given problem:
\[ x^2 - 9 = 0 \]
it's already in standard form.
Having the equation in standard form ensures you can easily identify the terms and use appropriate solving methods, like factoring, completing the square, or using the quadratic formula.

One of the simplest ways to solve a quadratic equation is by using the square root method. This method works particularly well when the equation can be easily transformed so that we can isolate the squared term. Given our exercise:
\[ x^2 - 9 = 0 \]
First, you need to isolate the quadratic term, \(x^2\). So, add 9 to both sides of the equation:
\[ x^2 = 9 \]
Next, take the square root of both sides. Remember that when you take the square root of both sides of an equation, you must consider both the positive and negative roots. This results in:
\[ x = \text{±}\textrm{\textbackslash}sqrt{9} \]
So,\( x = 3 \) or \( x = -3 \). This steps completes the solution using the square root method.

When solving quadratic equations, the solutions you find can be classified as rational, irrational, or complex. Rational solutions are numbers that can be expressed as the quotient of two integers. For our exercise, we found the solutions to be:
\[ x = 3 \quad \text{and} \quad x = -3 \]
Both 3 and -3 are integers and, thus, rational numbers.
Rational solutions are simpler to work with because they don't involve square roots of non-perfect squares or imaginary numbers.