This comprehensive guide covers all topics from the textbook chapter on quadratic equations. Each topic includes key concepts and memorization tips to help you master the material.
Definition: An equation of the form ax² + bx + c = 0 where a, b, c are real numbers and a ≠ 0.
Standard Form: ax² + bx + c = 0 (a ≠ 0)
Remember: “Quad” means square, so quadratic equations always have x² as the highest power. If a = 0, it’s not quadratic!
Three techniques:
Use factorization when you can easily spot factors. Use quadratic formula as a universal method. Completing squares is useful for deriving the formula.
Types:
For reciprocal equations, coefficients equidistant from start and end are equal. For radical equations, always check for extraneous roots!
Equation: x³ = 1 or x³ – 1 = 0
Roots: 1, ω, ω² where ω = (-1 + i√3)/2 and ω² = (-1 – i√3)/2
Properties: 1 + ω + ω² = 0, ω³ = 1, ω³ⁿ = 1
“One Omega Omega-squared” – Sum is zero, product is one. Each complex cube root is the square of the other.
Equation: x⁴ = 1 or x⁴ – 1 = 0
Roots: 1, -1, i, -i (two real, two imaginary)
Properties: Sum = 0, Product = -1
Four fourth roots: 1, -1, i, -i. i and -i are conjugates and multiplicative inverses. 1 and -1 are additive inverses.
Definition: P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀
Degree: Highest power of x (n)
Remainder Theorem: If P(x) ÷ (x-a), remainder = P(a)
Number of roots = Degree of equation. Graph of quadratic is parabola. Use Factor Theorem: (x-a) is factor if P(a)=0.
Quadratic: If α, β are roots of ax²+bx+c=0, then α+β = -b/a, αβ = c/a
Cubic: If α, β, γ are roots of ax³+bx²+cx+d=0, then α+β+γ = -b/a, αβ+βγ+γα = c/a, αβγ = -d/a
For sum of roots: coefficient of second term / first term with opposite sign. For product: constant term / first term.
Discriminant (Δ): Δ = b² – 4ac
Use discriminant to determine nature without solving! Positive = real, Zero = equal, Negative = imaginary.
Key properties:
Reciprocal roots → a=c, Opposite signs → b=0, Zero roots → c=0, Equal roots → Δ=0.