Algebraic Fractions and Equations
Algebraic Fractions and Quadratic Equations
This section deals with finding the solutions of equations such as
+ =
and uses techniques from the earlier sections.
Worked Examples
1
Find the solution of
+ =
Show me
The first stage is to combine the two fractions so that they have a common denominator.
| = | ||
| = | ||
| = | ||
Then dividing both sides by 2 gives | ||
| = | ||
Then multiplying both sides by 15 and (x − 6)(x + 2) gives | ||
| 15(x2 − 2x) | = (x − 6)(x + 2) | |
Therefore | ||
| 15x2 − 30x | = x2 − 4x − 12 | |
and | ||
| 14x2 − 26x + 12 | = 0 | |
| and dividing by 2 | 7x2 − 13x + 6 | = 0 |
This factorises as | ||
| (x − 1)(7x − 6) | = 0 | |
so
| x − 1 | = 0 | or | 7x − 6 | = 0 |
| x | = 1 | or | x = | = |
2
Solve the equation
+ =
Show me
The first step is to combine the two fractions using the ccmmon denominator x(x + 2).
| That is, | = | |
| = |
Then multiplying by 3 and x(x + 2) gives
| 3(x2 + x + 2) | = 5x(x + 2) |
| 3x2 + 3x + 6 | = 5x2 + 10x |
| 0 | = 2x2 + 7x − 6 |
This quadratic can be solved by using the formula (or by completing the square) to give
x =
where a = 2, b = 7 and c = −6,
giving
| x | = |
| = | |
| = | |
| x | = or |
Exercises
Solve each of the following equations.