Some math problems require the ability to calculate the square root. These problems include solving second-order equations. In this article, we present an effective method for calculating square roots and use it when working with formulas for the roots of a quadratic equation.
What is a square root?
In mathematics, this concept corresponds to the symbol √. Historical data says that it began to be used for the first time around the first half of the 16th century in Germany (the first German work on algebra by Christoph Rudolf). Scientists believe that this character is a transformed Latin letter r (radix means "root" in Latin).
The root of any number is equal to such a value, the square of which corresponds to the root expression. In the language of mathematics, this definition will look like this: √x=y if y2=x.
The root of a positive number (x > 0) is alsoa positive number (y > 0), but if the root of a negative number is taken (x < 0), then its result will already be a complex number, including the imaginary unit i.
Here are two simple examples:
√9=3 because 32 =9; √(-9)=3i because i2=-1.
Heron's iterative formula for finding square roots
The above examples are very simple, and calculating the roots in them is not difficult. Difficulties begin to appear already when finding the 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 roots for non-integer numbers: for example √(12, 15), √(8, 5) and so on.
In all the above cases, a special method of calculating the square root should be used. At present, several such methods are known: for example, expansion in a Taylor series, division by a column, and some others. Of all known methods, perhaps the simplest and most effective is the use of Heron's iterative formula, which is also known as the Babylonian method for determining square roots (there is evidence that the ancient Babylonians used it in their practical calculations).
Let it be necessary to determine the value of √x. The formula for finding the square root is as follows:
an+1=1/2(a+x/a), where limn->∞(a)=> x.
Decipher this mathematical notation. To calculate √x, you should take some number a0 (it can be arbitrary, but for a quick result, you should choose it such that (a0) 2 was as close as possible to x, then substitute it into the specified square root formula and get a new number a1, which will already be closer to the desired value. it is necessary to substitute a1 into the expression and get a2 This procedure should be repeated until the required accuracy is obtained.
An example of applying Heron's iterative formula
The algorithm described above for obtaining the square root of some given number may sound quite complicated and confusing for many, but in reality everything turns out to be much simpler, since this formula converges very quickly (especially if a lucky number is chosen a0).
Let's take a simple example: we need to calculate √11. We choose a0=3, since 32=9, which is closer to 11 than 42=16. Substituting into the formula, we get:
a1=1/2(3 + 11/3)=3, 333333;
a2 =1/2(3, 33333 + 11/3, 33333)=3, 316668;
a3=1/2(3, 316668 + 11/3, 316668)=3, 31662.
There is no point in continuing the calculations, since we have obtained that a2 and a3 begin to differ only in the 5th decimal place. Thus, it was enough to apply only 2 times the formula tocompute √11 to within 0.0001.
Currently, calculators and computers are widely used to calculate roots, however, it is useful to remember the marked 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 when solving quadratic equations. These equations are equalities with one unknown, the general form of which is shown in the figure below.
Here c, b and a are some numbers, and a must not be equal to zero, and the values of c and b can be completely arbitrary, including zero.
Any values of x that satisfy the equality indicated in the figure are called its roots (this concept should not be confused with the square root √). Since the equation under consideration has the 2nd order (x2), then there cannot be more than two roots for it. Let's look at how to find these roots later in the article.
Finding the roots of a quadratic equation (formula)
This method of solving the considered type of equalities is also called universal, or the method through the discriminant. It can be applied to any quadratic equations. The formula for the discriminant and roots of the quadratic equation is as follows:
It shows that the roots depend on the value of each of the three coefficients of the equation. Moreover, the calculationx1 differs from the calculation x2 only by the sign before the square root. The radical expression, which is equal to b2 - 4ac, is nothing but the discriminant of the considered equality. The discriminant in the formula for the roots of a quadratic equation plays an important role because it determines the number and type of solutions. So, if it is zero, then there will be only one solution, if it is positive, then the equation has two real roots, finally, the negative discriminant leads to two complex roots x1 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, Frenchman Francois Viet, studying second-order equations, was able to obtain the properties of its roots. Mathematically, they can be written like this:
x1 + x2=-b / a and x1 x 2=c / a.
Both equalities can easily be obtained by everyone, for this it is only necessary to perform the appropriate mathematical operations with the roots obtained through the formula with the discriminant.
The combination of these two expressions can rightfully be called the second formula of the roots of a 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 valid, it is convenient to use them to solve an equation only if it can be factored.
The task of consolidating the acquired knowledge
Let's solve a mathematical problem in which we will demonstrate all the techniques discussed in the article. The conditions of the problem are as follows: you need to find two numbers for which the product is -13, and the sum is 4.
This condition immediately reminds of Vieta's theorem, applying the formulas for the sum of square roots and their product, we write:
x1 + x2=-b / a=4;
x1 x2=c / a=-13.
Assuming a=1, then b=-4 and c=-13. These coefficients allow us to write a second order equation:
x2 - 4x - 13=0.
Use the formula with the discriminant, we get the following roots:
x1, 2=(4 ± √D)/2, D=16 - 41(-13)=68.
That is, the task was reduced to finding the number √68. Note that 68=417, then using the square root property, we get: √68=2√17.
Now let's use the considered square root formula: a0=4, then:
a1=1/2(4 + 17/4)=4, 125;
a2=1/2(4, 125 + 17/4, 125)=4, 1231.
There is no need to calculate a3 because the found values differ by only 0.02. Thus, √68=8.246. Substituting it into the formula for x 1, 2, we get:
x1=(4 + 8, 246)/2=6, 123 and x2=(4 - 8, 246) /2=-2, 123.
As you can see, the sum of the found numbers is really equal to 4, but if we find their product, then it will be equal to -12,999, which satisfies the condition of the problem with an accuracy of 0.001.
