The equation of type

**ах² + bx + c = 0**

where x i** s** a variable and

*d*

**a,**b**an**

*c**(a. 0),*real numbers are given is called

**a quadratic equation with one variable.**

The equation is

*a general*type of quadratic equation. If in it a

*= 1,*then it is in normal or reduced form and is written:

**x² + px + q = 0**

If in the equation * ax² + bx + c = 0 t*he coefficients a

*nd*

**,**b**a****are different from zero, then the equation is calle**

*c***d a solid quadratic equati**on, and if at least one of the coefficien

**s b or**

*t***c is equal to zero – an**

**incomplete quadratic equation.**

Incomplete quadratic equations are the equations:

**ax² + bx = **0,* c = 0**;*

**ax² + c = **0, b = 0*;*

##### 1. SOLVING INCOMPLETE SQUARE EQUATIONS

1. The equation of the* form ax² + bx = 0, *and ≠ 0, is solved by decomposing its left-hand side into factors, ie.

**x (ax + b) = 0**,

and it is equivalent to the set of equations:

It follows that the equation a* x² + bx = 0 *has two solutions:

so we say that* х₁ = х₂ = 0* is a double root of the equatio

**n aх² =**0.2. The equation of type

* ах² + c = 0, a ≠ 0*

is solved by shifting** **c to the right and dividing both sides

**y a, which gives:**

*b*– If * – c / a> *0, ie. when

*an*

**a***c have different signs, the equation has two different roots:*

**d**– If * – c / a *<0, ie. when

**an**

*a***c have equal signs, the roots of the equation are conjugate complex numbers:**

*d*##### 2. SOLVING A FULL SQUARE EQUATION

Solutions of the full quadratic equation

**ах² + bx + c = 0**

are determined by the formulas:

If the quadratic equation is given in normal form

then the formulas for the roots of the equation get the form:

This formula is suitable when the coefficient p *i*s an even number.

##### 3. DISCRIMINATION OF SQUARE EQUALITY

The real numbe**r b² – 4ac **is calle

*ant of the quadratic equat*

**d the discrimin***= 0, and is denoted b*

**ion ax² + bx + c****D, ie.**

*y***D = b² – 4ac**

Depending on the sign of D,** **the nature of the solutions of the quadratic equation aх² + bx + c = 0 (

*a*

**∀a, b, c ∈ R) and***≠ 0 can be d*eter

*mined.*

The solutions (roots) of the quadratic equation are:

1) real and different, if* D> 0*

2) real and equal, if *D = 0*

3) conjugate complex, if* D <0*

##### 4. VIET FORMULA

The numbers** x**₁ an

**х₂ are the roots of the quadratic equation**

*d***ах² + bx + c = 0**

if and only if the equations apply:

These equations are known as th**e Viet formulas**.

If the quadratic equation is given in normal form:

**х² + px + q = 0**,

then the Viet formulas are:

**х₁ + х₂ = -р **

**х₁ • х₂ = q**

##### 5. DISSOLUTION OF A SQUARE TRINOMA OF LINEAR MULTIPLE

The expre** ssion ах² + bx **+ c is call

**ed the quadratic tr**inomial with respect to the var

**able x, where**

*i**a, b, c ∈ R*an

*d a ≠*0.

The roots of the quadratic equation ** ax² + bx + c = 0 a**re z

*ero o*n the quadratic trinomial.

The quadratic trinom

**+ c can be decomposed into linear multipliers with real coefficients, ie.**

*ial ax² + bx**ax² + bx + c = a (x – x₁) (x – x₂)*