A rational algebraic equation can often be transformed into a quadratic equation by eliminating the denominators and rearranging the terms. Here's a general process and an example:

Process:

1. Identify the rational equation: A rational equation is an equation involving one or more rational expressions (fractions with polynomials in the numerator and/or denominator).

2. Clear the denominator: Multiply both sides of the equation by the least common denominator (LCD) to eliminate the fractions.

3. Simplify and rearrange: After clearing the fractions, simplify the resulting equation and rearrange it into the standard quadratic form $ax^2 + bx + c = 0$.

4. Solve the quadratic equation: Use the quadratic formula, factoring, or completing the square to solve the quadratic equation.

Example:

Consider the rational equation:

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

Step 1: Clear the denominators

• The denominators are $x$ and $x+1$.
• The LCD is $x(x+1)$.
• Multiply both sides of the equation by $x(x+1)$:

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

Step 2: Distribute and simplify

• Multiplying through gives:

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

• Simplify:

$x + 1 + 2x = 3x^2 + 3x$

$3x + 1 = 3x^2 + 3x$

Step 3: Rearrange into quadratic form

• Move all terms to one side of the equation to set it to zero:

$0 = 3x^2 + 3x - 3x - 1$

$0 = 3x^2 - 1$

• This simplifies to:

$3x^2 - 1 = 0$

Step 4: Solve the quadratic equation

• Add 1 to both sides:

$3x^2 = 1$

• Divide by 3:

$x^2 = \frac{1}{3}$

• Take the square root:

$x = \pm \frac{1}{\sqrt{3}} = \pm \frac{\sqrt{3}}{3}$

So, the solutions to the original rational equation are $x = \frac{\sqrt{3}}{3}$ and $x = -\frac{\sqrt{3}}{3}$.

Would you like more details or have any questions?

Related Questions:

1. Can you provide an example where the rational equation involves more complex denominators?
2. How would you solve a rational equation where the quadratic form does not factor easily?
3. What is the importance of checking for extraneous solutions in rational equations?
4. Can you explain how to use the quadratic formula on the derived quadratic equation?
5. How would you approach solving a rational equation where the quadratic equation has complex roots?

Tip: When transforming rational equations to quadratic form, always watch out for extraneous solutions by substituting your answers back into the original equation.