1/32
Vocabulary flashcards summarising key terms and definitions related to quadratics, inequalities, root conditions, and special elementary functions covered in the lecture notes.
Name | Mastery | Learn | Test | Matching | Spaced |
---|
No study sessions yet.
Quadratic Polynomial
An expression of the form f(x)=ax²+bx+c where a, b, c∈ℝ and a≠0.
Quadratic Equation
The statement ax²+bx+c=0 obtained by setting a quadratic polynomial equal to zero.
Leading Coefficient
The coefficient a of the highest-degree term in a polynomial (ax² for quadratics).
Absolute Term
The constant term c in ax²+bx+c.
Discriminant (D or Δ)
The quantity D=b²−4ac that determines the nature of the roots of ax²+bx+c=0.
Nature of Roots
Classification of roots based on D: D>0 real & distinct, D=0 real & equal, D<0 complex conjugate.
Perfect-Square Quadratic
A quadratic with D=0; it can be written as (px+q)²=0 and has one repeated real root.
Vertex of a Parabola
The turning point (−b/2a, −D/4a) of the graph y=ax²+bx+c.
Sum of Roots
For ax²+bx+c=0, α+β=−b/a.
Product of Roots
For ax²+bx+c=0, αβ=c/a.
Difference of Roots
|α−β|=√D / |a| for a quadratic.
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.’
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).
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).
Wavy-Curve Method
Graphical sign-analysis technique for solving polynomial inequalities: factorise, order critical points, mark signs, flip at odd multiplicity.
Modulus Function
|x|=x if x≥0, and −x if x<0; always non-negative.
Greatest Integer Function (GIF)
⌊x⌋ gives the largest integer ≤ x (also called the floor function).
Fractional Part Function (FPF)
{x}=x−⌊x⌋, the non-negative fractional portion of x, always in [0,1).
Rational Expression
A quotient h(x)=f(x)/g(x) where f and g are polynomials and g(x)≠0.
Domain
Set of all x for which a function is defined; for a polynomial it is ℝ.
Range
Set of y-values obtained from the function; for a quadratic it is bounded below (a>0) or above (a<0).
Root-Location Test (K-Test)
Evaluate f(K); if a·f(K)
Coefficient in A.P.
If a, b, c are in arithmetic progression, then 2b=a+c.
Coefficient in G.P.
If a, b, c are in geometric progression, then b²=ac.
Coefficient in H.P.
If a, b, c are in harmonic progression, then 2/a = 1/b + 1/c.
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).
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.
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.
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.
Special Condition 4a+2b+c=0
If 4a+2b+c=0, then x=2 is a root and the other root is c/(2a).
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.
Range of |x| Inequalities
|x|≤k gives x∈[−k,k]; |x|≥k gives x∈(−∞,−k]∪[k,∞) for k>0.
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.