Quadratic equations

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;

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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:

158-f1

It follows that the equation ax² + bx = 0 has two solutions:

159-f1

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:

159-f2

– If – c / a> 0, ie. when a and c have different signs, the equation has two different roots:

160-f1

– If – c / a <0, ie. when a and c have equal signs, the roots of the equation are conjugate complex numbers:

160-f2
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2. SOLVING A FULL SQUARE EQUATION

Solutions of the full quadratic equation

ах² + bx + c = 0

are determined by the formulas:

161-f2

If the quadratic equation is given in normal form

161-f3

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

162-f1

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

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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

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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:

164-f1

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

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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)

– square equations –


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