The solutions to quadratic equations can be rational or irrational. Recall that:

• Rational numbers can be written as positive or negative fractions. Numbers like 12, -3, , -4.79, and are rational. ( can be written as a fraction, because it’s equal to . The number -4.79 is the opposite of 4.79, which is .)
• Any number that is not rational is irrational. Some examples are , and . When an irrational number is written as a decimal, its digits go on forever without eventually making a repeating pattern, so a decimal can only approximate the value of the number.

How do we know if the solutions to a quadratic equation are rational or irrational?

If we solve a quadratic equation by graphing a corresponding function (), sometimes we can tell from the -coordinates of the -intercepts. Other times, we can't be sure.

Let's solve  and  by graphing  and .

Graphs of two functions on a grid. X axis from negative 4 to 2, by 2’s. Y axis from negative 8 to 4, by 4’s. Function y equals x squared minus 5 opens upward with points at negative 2 point 2 3 6 comma 0 and 2 point 2 3 6 comma 0. Vertex at 0 comma 5. Function y equals x squared minus 49 divided 100 opens upward with points at negative 0 point 7 comma 0 and 0 point 7 comma 0.

The graph of crosses the -axis at -0.7 and 0.7. There are no digits after 7, suggesting that the -values are exactly and , which are rational.

To verify that these numbers are exact solutions to the equation, we can see if they make the original equation true.

and , so are exact solutions.

The graph of , created using graphing technology, is shown to cross the -axis at -2.236 and 2.236. It is unclear if the -coordinates stop at three decimal places or if they continue. If they stop or eventually make a repeating pattern, the solutions are rational. If they never stop or make a repeating pattern, the solutions are irrational.

We can tell, though, that 2.236 is not an exact solution to the equation. Substituting 2.236 for in the original equation gives , which we can tell is close to 0 but is not exactly 0. This means are not exact solutions, and the solutions may be irrational.

To be certain whether the solutions are rational or irrational, we can solve the equations.

• The solutions to are , which are rational.
• The solutions to are , which are irrational. (2.236 is an approximation of , not equal to .)