Tip 1: How to solve the discriminant

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Tip 1: How to solve the discriminant

The solution of the quadratic equation is often reduced to finding discriminant. From its value depends on whether the equation has roots and how much it will be. Work around the search discriminant can only be done by the Viet formula theorem if the quadratic equation is reduced, that is, has a unit coefficient with the highest factor.

Instructions

Determine whether your equation is square. Such it will be, if it looks like: ax ^ 2 + bx + c = 0. Here a, b and c are numerical constant factors, and x is a variable. If the unit term is in the higher term (that is, the volume with the higher power, hence this is x ^ 2), then we can not look for the discriminant and find the roots of the equation by Viet's theorem, which says that the solution is the following: x1 + x2 = - b; x1 * x2 = c, where x1 and x2 are the roots of the equation, respectively. For example, the reduced quadratic equation: x ^ 2 + sx + 6 = 0. By Viet's theorem, a system of equations is obtained: x1 + x2 = -5; x1 * x2 = 6 . Thus, we obtain x1 = -2; x2 = -3.

If the equation is not reduced, then searches discriminant not to be avoided. Determine it by the formula: D = b ^ 2-4as. If the discriminant is less than zero, then the quadratic equation has no solutions, if the discriminant is zero, then the roots coincide, that is, the quadratic equation has only one solution. And only if the discriminant is strictly positive, the equation has two roots.

For example, the quadratic equation: 3x ^ 2-18x + 24 = 0.With the highest term there is a factor that is different from unity, hence, it is necessary to find the discriminant: D = 18 ^ 2-4 * 3 * 24 = 36. The discriminant is positive, therefore, the equation has two roots. X1 = (-b) + vD) / 2a = (18 + 6) / 6 = 4; x2 = (-b) -vD) / 2a = (18-6) / 6 = 2.

Complicate the problem by taking the following expression: 3x2 + 9 = 12x-x2. Carry all the terms to the left side of the equation, remembering to change the sign of the coefficients, and leave zero on the right-hand side: 3x ^ 2 + x ^ 2-12x + 9 = 0; 4x ^ 2 -12x + 9 = 0. Now, looking at this expression, we can say that it is square. Find the discriminant: D = (-12) ^ 2-4 * 4 * 9 = 144-144 = 0. The discriminant is zero, so this quadratic equation has only one root, which is determined by the simplified formula: x1,2 = -v / 2a = 12/8 = 3/2 = 1,5.

Tip 2: How to calculate the discriminant

To solve the quadratic equation, you must first determine its discriminant. Determined discriminant, we can immediately conclude about the number of roots of the quadratic equation. In the general case, to solve a polynomial of any order higher than the second, it is also necessary to look for discriminant.

You will need

mathematical operations

Instructions

Suppose you have a quadratic equation reduced to the form a (x * x) + b * x + c = 0. Its discriminant will be denoted by the letter D and will be equal to D = (b * b) -4ac.

The discriminant of the quadratic equation can be greater than zero, zero or less than zero. If it is greater than zero, then the equation has two real roots. If discriminant is equal to zero, then the equation has one real root. If discriminant is less than zero, then the equation has no realroots, but has two complex roots. The roots of the quadratic equation will be found by the formulas: x1 = (-b + sqrt (D)) / 2a, x2 = (-b-sqrt (D)) / 2a (in the case of real roots).

If the quadratic equation can be represented as a (x * x) + 2 * b * x + c = 0, then it is simpler to find the abbreviated discriminant of this equation in the form: D = (b * b) -ac. With such discriminantthe roots of the equation will look like this: x1 = (-b + sqrt (D)) / a, x2 = (-b-sqrt (D)) / a.

Tip 3: Square equations and how to solve them

A quadratic equation is a special kind of algebraicequation, whose name is due to the presence in it of a square term. Despite the seeming complexity, such equations have a clear algorithm for solving.

Quadratic equations and how to solve them

Equation that is squarea trinomial, is usually called a quadratic equation. From the point of view of algebra, it is described by the formula a * x ^ 2 + b * x + c = 0. In this formula, x is the unknown, which is to be found (it is called a free variable); a, b and c are numerical coefficients. There are a number of limitations with respect to the components of this formula: thus, the coefficient a must not be equal to 0.

Solution of the equation: the concept of discriminant

The value of the unknown x, for which the squarethe equation becomes a true equality, called the root of such an equation. In order to solve the quadratic equation, it is first necessary to find the value of the special coefficient - the discriminant, which will show the number of roots of the equality in question. The discriminant is calculated by the formula D = b ^ 2-4ac. The result of the calculation can be positive, negative or equal to zero. It should be borne in mind that the concept of a quadratic equation requires that only the coefficient a is strictly different from 0. Therefore, the coefficient b can be equal to 0, and the equation itself in this case is an example of the form a * x ^ 2 + c = 0. In this situation, you should use the coefficient value of 0, and in the formulas for calculating the discriminant and roots. So, the discriminant in this case will be calculated as D = -4ac.

The solution of the equation with a positive discriminant

In the case when the discriminant of the quadratic equationturned out to be positive, one can conclude from this that the given equation has two roots. The roots can be calculated by the following formula: x = (- b ± √ (b ^ 2-4ac)) / 2a = (- b ± √D) / 2a. Thus, in order to calculate the value of the roots of a quadratic equation with a positive discriminant value, the known values of the coefficients available in the equation are used. Due to the use of the sum and the difference in the formula for calculating the roots, the result of the calculations will be two values that turn this equality into the correct one.

Solution of the equation with zero and negative discriminant

In the case when the discriminant of the quadratic equationturned out to be equal to 0, we can conclude that this equation has one root. Strictly speaking, in this situation the roots of the equation are still two, but because of the zero discriminant they will be equal to each other. In this case, x = -b / 2a. If, in the course of calculations, the value of the discriminant turns out to be negative, it must be concluded that the quadratic equation in question does not have roots, that is, those values of x for which it turns into the correct equality.