Quadratic Equations, Inequalities & Special Functions – Vocabulary Review

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Vocabulary flashcards summarising key terms and definitions related to quadratics, inequalities, root conditions, and special elementary functions covered in the lecture notes.

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33 Terms

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Quadratic Polynomial

An expression of the form f(x)=ax²+bx+c where a, b, c∈ℝ and a≠0.

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Quadratic Equation

The statement ax²+bx+c=0 obtained by setting a quadratic polynomial equal to zero.

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Leading Coefficient

The coefficient a of the highest-degree term in a polynomial (ax² for quadratics).

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Absolute Term

The constant term c in ax²+bx+c.

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Discriminant (D or Δ)

The quantity D=b²−4ac that determines the nature of the roots of ax²+bx+c=0.

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Nature of Roots

Classification of roots based on D: D>0 real & distinct, D=0 real & equal, D<0 complex conjugate.

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Perfect-Square Quadratic

A quadratic with D=0; it can be written as (px+q)²=0 and has one repeated real root.

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Vertex of a Parabola

The turning point (−b/2a, −D/4a) of the graph y=ax²+bx+c.

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Sum of Roots

For ax²+bx+c=0, α+β=−b/a.

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Product of Roots

For ax²+bx+c=0, αβ=c/a.

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Difference of Roots

|α−β|=√D / |a| for a quadratic.

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Identity (Quadratic)

An expression ax²+bx+c that is 0 for all x; occurs only when a=b=c=0 and gives infinitely many ‘roots.’

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Condition for Infinite Roots

If a quadratic equals 0 for more than two x-values, it must be the zero polynomial (a=b=c=0).

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Common Root Condition

Two quadratics a₁x²+b₁x+c₁=0 and a₂x²+b₂x+c₂=0 share a root if (b₁c₂−b₂c₁)/(a₁c₂−a₂c₁)=(c₁a₂−c₂a₁)/(a₁b₂−a₂b₁).

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Wavy-Curve Method

Graphical sign-analysis technique for solving polynomial inequalities: factorise, order critical points, mark signs, flip at odd multiplicity.

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Modulus Function

|x|=x if x≥0, and −x if x<0; always non-negative.

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Greatest Integer Function (GIF)

⌊x⌋ gives the largest integer ≤ x (also called the floor function).

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Fractional Part Function (FPF)

{x}=x−⌊x⌋, the non-negative fractional portion of x, always in [0,1).

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Rational Expression

A quotient h(x)=f(x)/g(x) where f and g are polynomials and g(x)≠0.

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Domain

Set of all x for which a function is defined; for a polynomial it is ℝ.

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Range

Set of y-values obtained from the function; for a quadratic it is bounded below (a>0) or above (a<0).

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Root-Location Test (K-Test)

Evaluate f(K); if a·f(K)

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Coefficient in A.P.

If a, b, c are in arithmetic progression, then 2b=a+c.

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Coefficient in G.P.

If a, b, c are in geometric progression, then b²=ac.

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Coefficient in H.P.

If a, b, c are in harmonic progression, then 2/a = 1/b + 1/c.

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Transformation y=1/x

If α and β are roots of ax²+bx+c=0, then 1/α and 1/β are roots of cx²+bx+a=0 (swap a and c).

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Newton’s Theorem (GP Roots)

For roots in geometric progression p/r and pr of ax²+bx+c=0, the relation ap² + bp + c = 0 holds.

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Special Condition a+b+c=0

If a quadratic has a+b+c=0, then x=1 is a root and the other root is c/a.

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Special Condition a−b+c=0

If a quadratic has a−b+c=0, then x=−1 is a root and the other root is c/a.

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Special Condition 4a+2b+c=0

If 4a+2b+c=0, then x=2 is a root and the other root is c/(2a).

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Sign Rule for Roots

If a, b, c have the same sign, both roots are negative; if a and c share sign but b differs, both roots are positive; etc.

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Range of |x| Inequalities

|x|≤k gives x∈[−k,k]; |x|≥k gives x∈(−∞,−k]∪[k,∞) for k>0.

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Condition ‘Perfect-Square Discriminant’

If D is a non-zero perfect square and a,b,c are integers, roots are rational; if a=1 they are integers.