rational equation
an equation that contains one or more rational expressions
A rational equation is a type of algebraic equation that involves one or more rational expressions. A rational expression is a fraction where the numerator and/or denominator contain algebraic expressions, such as polynomials.
The general form of a rational equation is:
P(x) / Q(x) = R(x) / S(x)
where P(x), Q(x), R(x), and S(x) are polynomials, and x is the variable.
To solve a rational equation, we need to first find the values of x that make the equation true. To do this, we can multiply both sides of the equation by the common denominator of the two fractions and then simplify.
For example, consider the rational equation:
(5x + 2) / (x – 1) = (3x + 4) / (x + 2)
To solve this equation, we would start by multiplying both sides by the common denominator (x – 1)(x + 2):
(5x + 2)(x + 2) = (3x + 4)(x – 1)
Expanding and simplifying, we get:
5x^2 + 12x + 4 = 3x^2 + x – 4
Bringing all the terms to one side, we get:
2x^2 + 11x + 8 = 0
We can then factor this quadratic equation to get:
(2x + 1)(x + 8) = 0
This gives us two possible solutions: x = -1/2 or x = -8. However, we need to check that these solutions do not make any of the denominators zero. In this case, neither -1 nor -8 make any of the denominators zero, so they are both valid solutions to the original rational equation.
Therefore, the solutions to the rational equation (5x + 2) / (x – 1) = (3x + 4) / (x + 2) are x = -1/2 and x = -8.
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