Rational Equations Calculator with steps

Rational Equations Calculator

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Solving rational equations is very easy with the help of the Rational Equation Calculator. You just need to enter the equation with rational expressions into the calculator and press the Calculate button.

This calculator is an excellent tool for learning how to solve rational equations because it generates detailed step-by-step solutions.

In the next section we explain what rational equations are and how to solve them with examples.

What are rational equations?

Rational equations are all those equations that involve rational expressions. Or put in other words, a rational equation is an equation with at least one fraction.

Here is an example of a rational equation:

rational equation example 01
Fig. 1: Example of a rational equation

Rational equations can be used to model real-world situations where the relationship between two quantities is not a straight line, but rather a curve. For example, a rational equation might be used to model the relationship between the distance traveled and the time taken for a car traveling at a constant speed.

How to solve rational equations

Here are the steps to take to solve rational equations:

  1. Determine if there is any value of the variable that would cause any denominator to be zero.
  2. The next step is to determine the lowest common denominator of all the fractions in the equation.
  3. Then in order to eliminate the denominators, we multiply both sides of the equation by the lowest common denominator.
  4. Finally we solve the resulting equation and verify the solution. If any of the values from step 1 are found in the solution, you should remove them from the solution.

To show you how to apply these steps, we present a series of examples solved step by step. All the examples that we will present to you below were solved with the Rational Equation Calculator.

Rational equation examples

Example 01: 1x = 1x2−4x1x−4

1
x
=
1
x24x
1
x4
1
x
=
1
x(x4)
+
1
x4
Multiply all terms by x(x-4) and cancel:
1(x4)=11x
x4=x+1(Simplify both sides of the equation)
x4+x=x+1+x(Add x to both sides)
2x4=1
2x4+4=1+4(Add 4 to both sides)
2x=5
2x
2
=
5
2
(Divide both sides by 2)
x=
5
2
Check if the solution satisfies the equation
x=
5
2
(This value satisfy the equation)

Solution:
x=
5
2

Example 02: x+3x2−5 = 18

x+3
x25
=
1
8
Step 1: Multiply both sides by x^2-5.
x+3=
1
8
x2
+
5
8
x+3(
1
8
x2
+
5
8
)
=
1
8
x2
+
5
8
(
1
8
x2
+
5
8
)
(Subtract 1/8x^2+(-5)/8 from both sides)
1
8
x2
+x
+
29
8
=0
For this equation: a=-0.125, b=1, c=3.625
0.125x2+1x+3.625=0
x=
b±b24ac
2a
(Plug the coefficients into the quadratic formula)
x=
(1)±(1)24(0.125)(3.625)
2(0.125)
x=
1±2.8125
0.25
x=2.7082039324993694,10.70820393249937
Check if the solution satisfies the equation
x=2.708204(This value satisfy the equation)
x=10.708204(This value satisfy the equation)

Solution:
x=2.708204 or x=10.708204

Solving rational equations worksheet

We present a worksheet for you to practice with the help of the Rational Equations calculator.

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