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We always start by noting the values that would cause any denominators to be zero.
Solve:
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| Note any value of the variable that would make any denominator zero. |
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| Find the least common denominator of all denominators in the equation. The LCD is . | |
| Clear the fractions by multiplying both sides of the equation by the LCD. |
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| Distribute. |
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| Multiply. |
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| Solve the resulting equation. First write the quadratic equation in standard form. |
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| Factor. |
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| Use the Zero Product Property. |
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| Solve. |
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| Check. | |
| We did not get 0 as an algebraic solution. | |
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Solve:
 | |
| Note any value of the variable that would make any denominator zero. |
 |
| Find the least common denominator of all denominators in the equation. The LCD is . | |
| Clear the fractions by multiplying both sides of the equation by the LCD. |
 |
| Remove common factors. |
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| Simplify. |
 |
| Multiply. |
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| Solve the resulting equation. |
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| We did not get 0 or as algebraic solutions. | |
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When one of the denominators is a quadratic, remember to factor it first to find the LCD.
Solve:
 | |
| Note any value of the variable that would make any denominator zero. |
 |
| Find the least common denominator of all denominators in the equation. The LCD is . | |
| Clear the fractions by multiplying both sides of the equation by the LCD. |
 |
| Distribute. |
 |
| Remove common factors. |
 |
| Simplify. |
 |
| Distribute. |
 |
| Solve. |
 |
 | |
 | |
| We did not get as algebraic solutions. | |
 |
Solve:
 | |
| Note any value of the variable that would make any denominator zero. |
 |
| Find the least common denominator of all denominators in the equation. The LCD is . | |
| Clear the fractions by multiplying both sides of the equation by the LCD. |
 |
| Distribute. |
 |
| Remove common factors. |
 |
| Simplify. |
 |
| Simplify. |
 |
| Combine like terms. |
 |
| Solve. First write in standard form. |
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| Factor. |
 |
| Use the Zero Product Property. |
 |
| We did not get 4 or 3 as algebraic solutions. | |
 |
Solve:
 | |
| Factor all the denominators, so we can note any value of the variable the would make any denominator zero. |
 |
| Find the least common denominator of all denominators in the equation. The LCD is . | |
| Clear the fractions. |
 |
| Distribute. |
 |
| Remove common factors. |
 |
| Simplify. |
 |
| Solve the resulting equation. |
 |
 | |
| Check. The only algebraic solution was 4, but we said that 4 would make a denominator equal to zero. The algebraic solution is an extraneous solution. There is no solution to this equation. |
The equation we solved in [link] had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. Some equations have no solution.
Solve:
 | |
| Note any value of the variable that would make any denominator zero. |
 |
| Find the least common denominator of all denominators in the equation. The LCD is . | |
| Clear the fractions by multiplying both sides of the equation by the LCD. |
 |
| Distribute. |
 |
| Remove common factors. |
 |
| Simplify. |
 |
| Solve the resulting equation. |
 |
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 | |
 | |
| Check. | |
| is an extraneous solution. | |
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