Have you forgotten how to solve the incomplete quadratic equation?

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 real coefficients for the unknown x, and where a ≠ a, and b and c are zeros, either simultaneously or separately. For example, c = o, in ≠ o or vice versa. We almost remembered the definition of a 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 solutions of the equation can be complex numbers. It is customary to call the equation in which none of the coefficients is equal to a, and ≠ o, to ≠ o, with ≠ o.
Let's solve an example. 2х ² -9х-5 = о, we find
D = 81 + 40 = 121,
D is positive, then there are roots, x ₁ = (9 + √121): 4 = 5, and the second x ₂ = (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. 2х ² -9х-5 = 0 (ах ² + вх + с = о)

• We first find the discriminant D by the well-known formula in ² -4as.
• 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> o, the quadratic equation has only 2 distinct real roots, they are denoted by x ₁ usually and x ₂ ,
  Here's how to calculate:
  X ₁ = (-v + √D): (2a), and the second one: x ₂ = (-in-√D): (2a).
• D = o is one root, or, they say, two equal:
  X ₁ is x ₂ and is equal to-in: (2a).
• Finally, D

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

1. Ah ² + in x = o. The free term, the coefficient c for x ⁰ , is zero here, 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 x ₁ = 0, By calculations X ₂ = -b / a _.
2. Now the coefficient of x is equal to o, and c is not equal to (≠) o.
   X ² + c = o. We carry c to the right side of the equation, we get x ² = -c. This equation only has real roots only when -c is a positive number (c X _{1 is} then equal to √ (-c), respectively x ₂ - -√ (-c). Otherwise, the equation has no roots at all.
3. The last option: b = c = o, that is, ax ² = 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 full 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 = k ² - ac, the roots are calculated as follows: x _1,2 = (-k ± √ (D / 4)) / a for D> o.
  X = -k / a for D = o.
  There are no roots for D
• There are reduced square equations, when the coefficient of x in the square is 1, they are usually written x ² + px + q = o. All the above formulas apply to them, calculations are somewhat simpler.
  Example, x ² -4x-9 = 0. Calculate D: 2 ² +9, D = 13.
  X ₁ = 2 + √13, x ₂ = 2-√13.
• In addition, Viet's theorem is easily applied to the above . 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 x ₁ + x ₂ is -a / a, the product x ₁ · x ₂ is equal to c / a.

The sum of the free term c and the first coefficient a is 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. The coefficients can be in some relations among themselves

• X ² + x = o, 7 x ² -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.
  X ₁ = 1, x ₂ = 13/2.

There are a number of other ways to solve different second-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.