After studying the first-order equations in schools, the theme of square equalities is covered. There are several methods for solving them, however, the application of the discriminant formula is the most common and universal. We consider in this article this formula for solving quadratic equations.
What equations are called quadratic?
The figure below shows the equality consisting of three terms. The variable x is unknown. Since the first term contains it in the second degree, this expression is called square. Latin letters a, b and c in it indicate the numerical coefficients.
This equation is called complete because it contains all terms containing a variable in the 2nd, 1st, and 0th degrees (the term c, called free, can be represented as c * x 0 ).
If one of the coefficients b or c is zero, then the equation becomes incomplete. Note that the zero equality of a automatically converts the expression in question into a linear equation.
Both for complete and incomplete second-order equalities, one can use the formula for solving the quadratic equation in terms of discriminant.
Universal formula
As mentioned above, through the discriminant, the formula for solving the quadratic equation can be used to find the roots of second-order equality of absolutely any type. This formula is shown in the figure below.
It can be seen from it that the maximum equation can have two solutions (± sign), however, if the radical expression in the denominator is equal to zero, then the unknown x satisfying the equality will be represented by a single real number. The formula for solving the quadratic equation also demonstrates that its use is possible if all three (or less for an incomplete equation) of its coefficients are known.
The considered formula can be obtained independently, for this it is enough to solve the equation in general form using the method of complementing to a full square.
Note that this formula for determining the roots of incomplete equations does not need to be used, since there are simpler methods of solution (factorization by bracketing an x or simply transferring a free term to the right side of the equality and taking the root from it).
The concept of discriminant and its meaning
If we look again at the formula for solving the quadratic equation in terms of the discriminant, then the last difference will be the difference enclosed by the sign of the root in the denominator, that is, b 2 - 4 * a * c.
What role does he play? Not knowing anything about the equation, but having only its discriminant, we can confidently say how many solutions it has and what type they are. So, a positive value of the discriminant corresponds to 2 real solutions, a negative value of it also speaks of 2 solutions, but they will already be complex numbers. Finally, if the discriminant is zero, which is true when b * b = 4 * a * c, then the equation will have only one real root x.
Examples of solving second-order equalities
Using the formula of the roots of the quadratic equation, we give the solution of quadratic equations in problems of a different nature.
Problem number 1. The product of some 2 numbers is -84, and their sum is 5. These numbers need to be determined.
We compose a system of equations according to a given condition, we obtain:
x 1 * x 2 = -84
x 1 + x 2 = 5
Express from the second equation x 1 , substitute it in the first:
(5 - x 2 ) * x 2 = -84 = - (x 2 ) 2 + 5 * x 2
Now you should transfer the terms with x and x in the square to the left side and calculate the discriminant:
(x 2 ) 2 - 5 * x 2 - 84 = 0; D = 25 - 4 * 1 * (-84) = 361
Using the universal formula, we obtain the value of the roots of the equation:
x 2 = (5 ± 19) / 2 => x 2 = (12; -7)
To get x 1 , you can use any of the equations of the system. Substituting the known values of x 2 , we obtain similar numbers for x 1 . This fact means that only one pair of numbers, that is -7 and 12, satisfies the condition of the problem.
Problem number 2. Now we will solve a somewhat unusual problem. The following is the equation:
x 2 - k * x + 36 = 0
It is necessary to find all values of k that would lead to a unique solution to the equality.
To understand how to answer the question posed, it should be remembered that equations of the type in question have 1 root only if its discriminant is zero. That is, we need to find this discriminant, where can we get the number k. We have:
D = k 2 - 4 * 1 * 36 = 0
The resulting equality is called a pure second-order equation (there is no coefficient b in it). We solve it:
k = ± √144 = ± 12
Thus, if the number k takes the value +12 or -12, then the root of the equation will be one.