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Rational Equations, Pythagorean Theorem and Quadratic Equations

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1. Explain the five-steps for solving rational equations. Can any of these steps be eliminated? Can the order of these steps be changed? Would you add any steps to make it easier, or to make it easier to understand? Provide an example with your explanation.

2. How might the Pythagorean theorem and distance formula be used in everyday life? Provide examples of each. Show how the formula is used in the example you provide.

3. Quadratic equations, which are expressed in the form of ax2 + bx + c = 0, where a does not equal 0, may have how many solutions? Explain why and prove with an example.

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1.) Answer: The 5 steps for solving rational equations are as follows:

- Determine the least common denominator (LCD) of all the fractions in the given rational equation.
- Multiply each terms in both sides of the rational equation by the LCD.
- Simplify the equation by combining the common terms.
- Solve the rational equation with the use of mathematical manipulation of the terms.
- Check the solution if it is valid by substituting it to the original equation and determine if the rational equation is valid after substitution.

An additional step, or a substitutional step, that can be done, especially if both sides of the equation are a single-term rational function, is to use cross-multiplication. This step solves the ...

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