Rooting Out Results from Quadratic Equations

The general quadratic equation has the form ax2 + bx + c = 0, and b or c or both of them can be equal to 0. This section shows you how nice it is — and how easy it is to solve equations — when b is equal to 0. The first 20 perfect squares (products of a number times itself) are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, and 400.

Notice that the square numbers go from a low of 1 to a high of 400. There aren’t any other perfect squares between the ones listed. That means that the other 380 numbers between 1 and 400 are not perfect squares. The perfect squares all have nice square roots. The square root of 121 is 11; the square root of 256 is 16. Isn’t that nice?

But the square root of 200 isn’t nice at all; it’s an irrational number. Irrational numbers don’t terminate or repeat themselves after the decimal point. For example, the square root of 2, an irrational number, is 1.414213562373. . . . An irrational number can’t ever be written as a fraction. Irrationals are just as their name describes: wild and unpredictable. The roots have decimal values that can be approximated with a calculator, though.


Don’t worry if you don’t recognize some of the larger squares because they aren’t used frequently, and you usually get some sort of a hint that the number is a perfect square when you’re doing a problem. Sometimes the hint comes from the wording of the problem — it may talk about a square room or sides of a right triangle. Sometimes the hint is just that it’d be so nice if it were square.

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