How to solve an incomplete quadratic equation? It is known that it is a particular variant of the equality ax2+ bx + c = a, where a, b and c are realcoefficients for unknown x, and where a ≠ a, and b and c are zeros-simultaneously or separately. For example, c = o, in ≠ o or vice versa. We almost recalled the definition of the quadratic equation.

We will clarify

The trinomial of the second degree is equal to zero. Its first coefficient a ≠ o, b and c can take any values. The value of the variable x will then be the root of the equation, when substituting it, will return it to the correct numerical equality. Let us dwell on real roots, although the solution of the equation can be complex numbers. It is customary to call an equation in which none of the coefficients is equal to a, and ≠ o, to ≠ o, c ≠ o.

Let's solve an example. 2x2-9x-5 = 0, we find

D = 81 + 40 = 121,

D is positive, then there are roots, x1= (9 + √121): 4 = 5, and the second x2= (9-√121): 4 = -o, 5. Checking will help to make sure that they are correct.

Here is a step-by-step solution of the quadratic equation

Through the discriminant, any equation can be solved, on the left side of which there is a known quadratic trinomial for a ≠ o. In our example. 2x2-9x-5 = 0 (ax2+ вх + с = о)

- We first find the discriminant D from the well-known formula in2-4ас.
- We check what the value of D will be: we have more than zero, it is equal to zero or less.
- We know that if D> 0, the quadratic equation has only 2 distinct real roots, they are denoted by x1usually x2,

Here's how to calculate:

x1= (-B + √D): (2a), and the second: x2= (-in-√D): (2a). - D = o is one root, or, they say, two equal:

x1is equal to x2and is equal to: (2a). - Finally, D <o means that the equation has no real roots.

Let us consider what are the incomplete equations of the second degree

- Oh2+ ix = o. The free term, the coefficient c for x0, here is zero, in ≠ o.

How to solve an incomplete quadratic equation of this kind? We take x for the brackets. We recall when the product of two factors is zero.

x (ax + b) = o, this can be when x = 0 or when ax + b = o.

Solving the second linear equation, we have x = -v / a.

As a result, we have the roots x1= 0,by calculationsx2= -b / a. - Now the coefficient of x is equal to o, and c is not equal to (≠) o.

x2+ c = o. We transfer c from the right-hand side of the equality, we obtain x2= -c. This equation only has real roots only when -c is a positive number (c <o),

x1then is equal to √ (-c), respectively, x2- -√ (-s). Otherwise, the equation has no roots at all. - The last option: b = c = o, that is, ah2= o. Naturally, such a simple equation has one root, x = o.

Special cases

How to solve the incomplete quadratic equation considered, and now we take any kinds.

- In the complete quadratic equation, the second coefficient for x is an even number.

Let k = o, 5b. We have formulas for calculating the discriminant and roots.

D / 4 = k2- ac, the roots are computed as x1.2= (-k ± √ (D / 4)) / a for D> o.

x = -k / a for D = o.

There are no roots for D <o. - There are reduced square equations, when the coefficient of x in the square is 1, they are customarily written x2+ px + q = o. All the above formulas apply to them, but calculations are somewhat simpler.

Example, x2-4x-9 = 0. We compute D: 22+9, D = 13.

x1= 2 + √13, x2= 2-√13. - In addition, the above is easily appliedViet's theorem. It says that the sum of the roots of the equation is -p, the second coefficient with the minus sign (meaning the opposite sign), and the product of these same roots is equal to q, the free term. Check how easy it would be to verbally determine the roots of this equation. For unreduced (for all coefficients not equal to zero) this theorem is applicable as follows: the sum x1+ x2is equal to -a / a, the product x1· X2is equal to c / a.

The sum of the free term c and the first coefficient ais equal to the coefficient b. In this situation, the equation has at least one root (easy to prove), the first must be -1, and the second must be c / a, if it exists. How to solve the incomplete quadratic equation, you can check yourself. As easy as pie. Coefficients can be in some relations among themselves

- x2+ x = o, 7x2-7 = o.
- The sum of all the coefficients is o.

The roots of this equation are 1 and c / a. Example, 2x2-15x + 13 = o.

x1= 1, x2= 13/2.

There are a number of other ways to solve differentsecond-degree equations. Here, for example, is the method of separating a complete square from a given polynomial. There are several graphic ways. When you often deal with such examples, you will learn how to "click" them like seeds, because all the ways come to mind automatically.

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