The solution of an algebraic equation, according to a larger account, comes down to finding its roots. The calculation of the discriminant of a given expression will not only find out the number of solutions of the equation (roots), but also determine their belonging to a real or complex numerical set. Most often, the term discriminant is used when working with square equations.
Discriminant - What is it?
The term "discriminant" is inextricably linked with the concept of polynomial - the expression
p (β) \u003d a.0*β n.+ a.1*β n-1+ a.2*β n-2.+ … + a.n-1*β + a.n., where
β - unknown variable,
a.n., a.n-1, a.n-2., … a.1 and a.0 - Numeric constants (constants).
That. Discriminant of polynomial P (β) with roots β 1, β 2 … β n.it is a product of species a.0 2N-2.∏(β i. – β j.)2, with I \u003cJ.
Denotes this characteristic letter D: D (β) \u003d a.0 2N-2.∏(β i. – β j.)2.
Discriminant of second-order equations
Most often, the concept of "discriminant" is used when working with square equations. The equation of the second degree (or the square equation) is an expression, the maximum erection of a variable in which is equal to 2.
General view: A * M 2.+ b * m + c \u003d 0, where:
a, b, c - numerical constants,
m is an unknown variable.
If all 3 terms are present, they say that the equation is complete. If any of the members is absent, in front of you, according to the incomplete equation of degree 2.
The discriminant in this case represents a certain auxiliary value, which allows not only to establish the number of solutions of the equation, but also to unambiguously determine their value. Based on the ratios in the formula for finding the discriminant of the N-order equation, the desired expression is transformed as follows:
D \u003d B. 2 - 4 A * C, where:
- a - the numerical constant before the variable in the senior (2nd) degree,
- b - a constant numerical expression before the first degree variable,
- c is a free member of the equation.
The relationship of discriminant and the roots of the square equation
To find the roots of the second order equation, the following ratio will be fair:
m. 1,2 \u003d (-b ± √d) / 2a, where
m. 1,2- solutions of a square equation.
From this ratio it is easy to see that:
- If the value of the discriminant is a value of positive (d\u003e 0), the equation has 2 different values \u200b\u200bof real root.
- If the discriminant has a negative value (D \u003c0), the equation also has 2 different solutions, but they are already among many complex numbers.
- If the size of the discriminant is identical to zero (d \u003d 0), the expression has 2 equal solutions among themselves.
Definition of discriminant - physical meaning
The relationship of the number of solutions of the second-order equation and the size of the discriminant also has a graphical justification. Physically the essence of the solution of the square equation is to fix zeros of parabola (intersection points with the abscissa axis), which it specifies. Visually this relationship illustrates the images below.
347
