Quadratic Inequation – Inequality Methods in Algebra :: Danielitte

Convex Quadrilateral

Segment of Circle

Sector of Circle

Regular Poligon of N Sides

Sperical Segment

Sperical Sector

Frustum of Right Circular Cone

Triangular Prism

Arithmetic Progression

Geometric Progression

Decimal Logarithm

Natural Logarithm

Euler's Formula

Trigonometric Functions For A Right Triangle

Trigonometric Table

Multiple Angle Formulas

Powers Of Trigonometric Functions

Formulas With T=tan(x/2)

Addition Formulas

Sum Of Trigonometric Functions

Product Of Trigonometric Functions

Half Angle Formulas

Angles Of A Plane Triangle

Sides And Angles Of A Plane Triangle

Relationships Among Trigonometric Fuctions

Linear Equation

System of Two Linear Equation

Quadratic Equation

Exponential Equation

Logarithmic Equation

Trigonometric Equation Cos

Trigonometric Equation Sin

Trigonometric Equation Tan

Trigonometric Equation Cotan

Linear Inequation

Quadratic Inequation

Exponential Inequation

Logarithmic Inequation

Trigonometric Inequation Cos

Trigonometric Inequation Sin

Trigonometric Inequation Tan

Trigonometric Inequation Cotan

Horizontal Shifting

Vertical Shifting

Equation of Line

Equation of Circle

Equation of Line Joining Two Points A,B

Equation of Sphere Center at M and Radius R in Rectangular Coordinates

Equation of Ellipsoid With Center M and Semi-Axes A,B,C

Elliptic Cylinder With Axis as Z Axis

Elliptic Cone With Axis as Z Axis

Hyperboloid of One Sheets

Hyperboloid of Two Sheets

Elliptic Paraboloid

Hyperbolic Paraboloid

Differentiation

Indefinite Integrals

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Integrals Involving Roots

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Relative Complement of A in B

Absolute Complement

Symmetric Difference

Operations On Sets

Standard Deviation

Root Mean Square

Normal Distribution(gaussion Distribution)

Exponential Distribution

Poisson Distribution

Uniforn Distribution

Real Form Of Fourier Series

Parseval's Theorem

Fourier Transform

Fourier Symmetry Relationships

Fourier Transform Pairs

Substitution(Frequency Shifting)

Translation(Time Shifting)

Laplace Transform Pairs

Equations - Quadratic Inequation

Quadratic Inequation

General Form:

\[ ax^2 + bx + c > 0 \quad \text{where } a \neq 0 \]

Terminology:

\( a \): Coefficient of \( x^2 \), determines the direction of the parabola (opens up if \( a > 0 \), down if \( a < 0 \)).
\( b \): Coefficient of \( x \).
\( c \): Constant term.
Discriminant (\( \Delta \)): \( b^2 - 4ac \), determines the number and type of roots.
Roots: Values of \( x \) where \( ax^2 + bx + c = 0 \), denoted \( x_1 \) and \( x_2 \).

Solution Cases:

Case 1: If \( a > 0 \)

\( b^2 - 4ac < 0 \): The quadratic expression is always positive ⇒ inequality holds for all real \( x \).
\( b^2 - 4ac = 0 \): Expression is positive for all \( x \ne -\frac{b}{2a} \).
\( b^2 - 4ac > 0 \): Expression is positive outside the interval between roots ⇒ \( x < x_1 \) or \( x > x_2 \).

Case 2: If \( a < 0 \)

\( b^2 - 4ac \leq 0 \): Expression never positive ⇒ no real solution.
\( b^2 - 4ac > 0 \): Expression is positive between the roots ⇒ \( x_1 < x < x_2 \).

Quadratic Formula (for finding roots):

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Applications:

Physics: Used in projectile motion and kinematic equations.
Engineering: In designing curves like bridges, arches, and lenses.
Economics: For analyzing profit, loss, and cost curves.
Computer Science: In algorithm optimization and graphics rendering.
Mathematics: Solving inequalities and graphing parabolas.

Quadratic Inequation