Some problems in mathematics require the ability to calculate the value of the square root. Such tasks include solving second-order equations. In this article, we present an effective method for calculating square roots and use it when working with the root formulas of the quadratic equation.
What is the square root?
In mathematics, the symbol √ corresponds to this concept. Historical evidence suggests that it began to be used for the first time around the first half of the 16th century in Germany (the first German algebra work by Christoph Rudolph). Scientists believe that this symbol is a transformed Latin letter r (radix means "root" in Latin).
The root of a number is equal to a value whose square corresponds to the radical expression. In the language of mathematics, this definition will look like this: √x = y, if y 2 = x.
A root from a positive number (x> 0) is also a positive number (y> 0), however, if a root from a negative number (x <0) is taken, then its result will be a complex number including the imaginary unit i.
Here are two simple examples:
√9 = 3, since 3 2 = 9; √ (-9) = 3i, since i 2 = -1.
Heron's Iterative Formula for Finding the Values of Square Roots
The above examples are very simple, and calculating the roots in them is not difficult. Difficulties begin to appear even when finding root values for any value that cannot be represented as a square of a natural number, for example, √10, √11, √12, √13, not to mention the fact that in practice it is necessary to find the roots for non-integer numbers: for example √ (12.15), √ (8.5), and so on.
In all the above cases, a special method for calculating the square root should be used. Currently, there are several such methods: for example, Taylor series expansion, column division, and some others. Of all the known methods, perhaps the simplest and most effective is the use of the Heron iterative formula, which is also known as the Babylonian method of determining square roots (there is evidence that the ancient Babylonians used it in their practical calculations).
Let it be necessary to determine the value √x. The formula for finding the square root is as follows:
a n + 1 = 1/2 (a n + x / a n ), where lim n-> ∞ (a n ) => x.
Decipher this mathematical notation. To calculate √x, one should take a number a 0 (it can be arbitrary, however, to quickly get the result, choose it so that (a 0 ) 2 is as close to x as possible. Then substitute it into the specified square root formula and get a new one the number a 1 , which will already be closer to the desired value, after which it is necessary to substitute a 1 into the expression and get a 2 .
An example of the application of the iteration formula of Heron
The algorithm for obtaining the square root of a given number described above may sound complicated and confusing for many, but in reality it turns out to be much simpler, since this formula converges very quickly (especially if a good number a 0 is chosen).
We give a simple example: it is necessary to calculate √11. We choose a 0 = 3, since 3 2 = 9, which is closer to 11 than 4 2 = 16. Substituting into the formula, we get:
a 1 = 1/2 (3 + 11/3) = 3.333333;
a 2 = 1/2 (3.33333 + 11 / 3.33333) = 3.316668;
a 3 = 1/2 (3.316668 + 11 / 3.316668) = 3.31662.
Further, it makes no sense to continue the calculations, since we have obtained that a 2 and a 3 begin to differ only in the 5th decimal place. Thus, it was enough to apply the formula only 2 times to calculate √11 with an accuracy of 0.0001.
Nowadays, calculators and computers are widely used for calculating roots, however, it is useful to remember the noted formula in order to be able to manually calculate their exact value.
Second order equations
Understanding what a square root is and the ability to calculate it is used in solving quadratic equations. Equations with one unknown are called these equations, the general form of which is shown in the figure below.
Here c, b and a are some numbers, moreover, a should not be zero, and the values of c and b can be completely arbitrary, including zero.
Any x values satisfying the equality indicated in the figure are called its roots (this concept should not be confused with the square root √). Since the equation in question is of the second order (x 2 ), there can be no more roots for it than two numbers. We will consider further in the article how to find these roots.
Finding the roots of the quadratic equation (formula)
This method of solving the considered type of equalities is also called universal, or the method through discriminant. It can be used for any quadratic equations. The formula of the discriminant and the roots of the quadratic equation has the following form:
It can be seen from it that the roots depend on the value of each of the three coefficients of the equation. Moreover, the calculation of x 1 differs from the calculation of x 2 only in the sign in front of the square root. The radical expression, which is equal to b 2 - 4ac, is nothing more than the discriminant of the equality under consideration. The discriminant in the formula of the roots of the quadratic equation plays an important role, since it determines the number and type of solutions. So, if it is equal to zero, then there will be only one solution, if it is positive, then the equation has two real roots, and finally, the negative discriminant leads to two complex roots x 1 and x 2 .
Vieta's theorem or some properties of the roots of second-order equations
At the end of the 16th century, one of the founders of modern algebra, the Frenchman François Viet, studying second-order equations, was able to obtain the properties of its roots. Mathematically, they can be written as follows:
x 1 + x 2 = -b / a and x 1 * x 2 = c / a.
Both equalities can be easily obtained by everyone, for this it is only necessary to perform the corresponding mathematical operations with the roots obtained through the formula with discriminant.
The combination of these two expressions can rightfully be called the second formula of the roots of the quadratic equation, which makes it possible to guess its solutions without using the discriminant. It should be noted here that although both expressions are always true, it is convenient to use them to solve the equation only if it can be factorized.
The task of consolidating the knowledge gained
We solve a mathematical problem in which we demonstrate all the techniques discussed in the article. The conditions of the problem are as follows: it is necessary to find two numbers for which the product is -13 and the sum is 4.
This condition immediately recalls the Vieta theorem, using the formula for the sum of square roots and their products, we write:
x 1 + x 2 = -b / a = 4;
x 1 * x 2 = c / a = -13.
If we assume that a = 1, then b = -4 and c = -13. These coefficients allow you to create a second order equation:
x 2 - 4x - 13 = 0.
We use the discriminant formula, we get the following roots:
x 1,2 = (4 ± √D) / 2, D = 16 - 4 * 1 * (-13) = 68.
That is, the problem was reduced to finding the number √68. Note that 68 = 4 * 17, then, using the square root property, we get: √68 = 2√17.
Now we use the considered square root formula: a 0 = 4, then:
a 1 = 1/2 (4 + 17/4) = 4.125;
a 2 = 1/2 (4.125 + 17 / 4.125) = 4.1231.
There is no need to calculate a 3 , since the values found differ by only 0.02. Thus, √68 = 8.246. Substituting it in the formula for x 1,2 , we get:
x 1 = (4 + 8.246) / 2 = 6.123 and x 2 = (4 - 8.246) / 2 = -2.123.
As you can see, the sum of the numbers found is really 4, but if you find their product, it will be -12,999, which satisfies the problem condition with an accuracy of 0.001.