91Ó°ÊÓ

When solving rational equations, it's crucial to identify excluded values. These are the numbers that make the denominator equal to zero. Since dividing by zero is undefined, these values cannot be part of the solution.
To find excluded values:

• Examine the denominators in the rational equation.
• Set the denominator equal to zero, as these are the numbers that would make the equation undefined.
• Solve the resulting equation to find excluded values.

In the equation \(x - \frac{10}{x} = -3\), the denominator is \(x\). Setting \(x = 0\) shows that zero is an excluded value because it would make the expression \(\frac{10}{x}\) undefined.

The denominator of a fraction is the part below the division line. It tells us how many equal parts the numerator is divided into. In rational equations, the denominator plays a key role since it can affect which values of the variable are valid solutions.
By identifying a common denominator, we can often simplify rational expressions. This step can help eliminate fractions from the equation, simplifying the solving process.
In this exercise, the only denominator present is \(x\). Multiplying the entire equation by \(x\) clears the denominator, transforming the equation to the quadratic form \(x^2 - 10 = -3x\). This step is crucial as it allows us to work with a simpler polynomial equation without fractions.

A quadratic equation is a polynomial equation of degree 2, typically written as \(ax^2 + bx + c = 0\). These equations can have up to two solutions, found using various methods such as factoring, completing the square, or the quadratic formula.
After clearing the fraction in our exercise example, we rearrange the terms to form the quadratic equation \(x^2 + 3x - 10 = 0\). This involves grouping all terms on one side of the equation. Quadratic equations are pivotal because they help us find the possible values of \(x\) that satisfy the equation.

Factoring is a method used to simplify mathematical expressions and solve equations, commonly applied to quadratic equations. It involves breaking down a complex expression into simpler factors or terms.
To factor a quadratic like \(x^2 + 3x - 10 = 0\), we search for two numbers that multiply to the constant term, -10, and add to the linear coefficient, 3. These numbers are 5 and -2. Thus, the equation factors into \((x + 5)(x - 2) = 0\).
By setting each factor equal to zero, we solve the equation by isolating the variable \(x\). This results in the solutions \(x = -5\) and \(x = 2\). Factoring transforms the quadratic equation into a form that is much easier to solve.