The equation of type
ах² + bx + c = 0
where x is a variable and a, b and 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 the coefficients a, b and c are different from zero, then the equation is called a solid quadratic equation, and if at least one of the coefficients b or 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 ax² + bx = 0 has two solutions:
so we say that х₁ = х₂ = 0 is a double root of the equation aх² = 0.
2. The equation of type
ах² + c = 0, a ≠ 0
is solved by shifting c to the right and dividing both sides by a, which gives:
– If – c / a> 0, ie. when a and c have different signs, the equation has two different roots:
– If – c / a <0, ie. when a and c have equal signs, the roots of the equation are conjugate complex numbers:
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 is an even number.
3. DISCRIMINATION OF SQUARE EQUALITY
The real number b² – 4ac is called the discriminant of the quadratic equation ax² + bx + c = 0, and is denoted by D, ie.
D = b² – 4ac
Depending on the sign of D, the nature of the solutions of the quadratic equation aх² + bx + c = 0 (∀a, b, c ∈ R) and a ≠ 0 can be determined.
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 x₁ and х₂ are the roots of the quadratic equation
ах² + bx + c = 0
if and only if the equations apply:
These equations are known as the 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 expression ах² + bx + c is called the quadratic trinomial with respect to the variable x, where a, b, c ∈ R and a ≠ 0.
The roots of the quadratic equation ax² + bx + c = 0 are zero on the quadratic trinomial.
The quadratic trinomial ax² + bx + c can be decomposed into linear multipliers with real coefficients, ie.
ax² + bx + c = a (x – x₁) (x – x₂)
