A rational algebraic equation is an equation involving rational expressions, which are fractions with polynomials in the numerator and denominator. Sometimes, these equations can be transformed into quadratic equations by clearing denominators and simplifying.

Steps to Transform a Rational Equation into a Quadratic Equation

1. Identify the Equation: Start with a rational equation of the form: $\frac{P(x)}{Q(x)} = \frac{R(x)}{S(x)}$ where $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ are polynomials.

2. Clear the Denominators: Multiply both sides of the equation by the least common multiple (LCM) of the denominators $Q(x)$ and $S(x)$ to eliminate fractions: $P(x) \cdot S(x) = R(x) \cdot Q(x).$ This will yield a polynomial equation.

3. Simplify to Obtain a Quadratic Form: Expand and simplify the resulting polynomial equation. Often, you'll end up with a quadratic equation in the form: $ax^2 + bx + c = 0,$ where $a$, $b$, and $c$ are constants.

4. Solve the Quadratic Equation: Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions for $x$.

Example Problem

Let's consider the following rational equation:

$\frac{2}{x} + 3 = \frac{x + 1}{x - 1}.$

Step-by-Step Transformation

1. Clear the Denominators: The LCM of the denominators $x$ and $x - 1$ is $x(x - 1)$. Multiply both sides by $x(x - 1)$:

   $x(x - 1) \left( \frac{2}{x} + 3 \right) = x(x - 1) \left( \frac{x + 1}{x - 1} \right).$

   This simplifies to:

   $2(x - 1) + 3x(x - 1) = x(x + 1).$

2. Expand and Simplify: Distribute and combine like terms:

   $2x - 2 + 3x^2 - 3x = x^2 + x.$

   Combine all terms into a standard quadratic form:

   $3x^2 - 2x - 2 = x^2 + x.$

   Subtract $x^2 + x$ from both sides:

   $2x^2 - 3x - 2 = 0.$

3. Solve the Quadratic Equation: Use the quadratic formula where $a = 2$, $b = -3$, and $c = -2$:

   $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2}.$

   Simplify:

   $x = \frac{3 \pm \sqrt{9 + 16}}{4} = \frac{3 \pm \sqrt{25}}{4}.$

   This gives:

   $x = \frac{3 + 5}{4} = 2, \quad x = \frac{3 - 5}{4} = -\frac{1}{2}.$

Solution

The solutions to the original rational equation are $x = 2$ and $x = -\frac{1}{2}$. However, check that these solutions do not make the original denominators zero (in this case, neither does, so both are valid).

Would you like more examples, or have any specific questions?

Related Questions You Might Find Interesting:

1. How can we solve a rational equation that involves more complex fractions?
2. What are the steps to graph a quadratic function?
3. How do you identify extraneous solutions when solving rational equations?
4. What are the different methods for solving quadratic equations (factoring, completing the square, etc.)?
5. Can we apply the quadratic formula to equations that do not initially look quadratic?

Tip: Always check for extraneous solutions when solving rational equations by substituting back into the original equation!