Quadratic Equations — NDA Maths

Formulas (8)

• What a Quadratic Equation Is · Standard form
  $$ax^2 + bx + c = 0, \quad a \neq 0$$
• Three Ways to Solve a Quadratic · Quadratic formula
  $$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
• The Discriminant — Nature of the Roots · Discriminant
  $$D = b^2 - 4ac$$
• Equal Roots Force a Coefficient Progression · Progression tests
  $$\text{GP}:\ b^2 = ac \qquad \text{AP}:\ 2b = a+c \qquad \text{HP}:\ \tfrac{2}{b} = \tfrac{1}{a}+\tfrac{1}{c}$$
• Difference and Ratio of the Roots · Difference of roots
  $$|\alpha - \beta| = \sqrt{(\alpha+\beta)^2 - 4\alpha\beta} = \dfrac{\sqrt{D}}{|a|}$$
• The a + b + c = 0 Shortcut · Unit-root test
  $$a + b + c = 0 \iff x = 1 \text{ is a root},\quad \text{other root} = \tfrac{c}{a}$$
• Location of the Roots in an Interval · Both roots in (p, q), a > 0
  $$D \ge 0,\quad f(p) > 0,\quad f(q) > 0,\quad p < -\tfrac{b}{2a} < q$$
• Equations That Reduce to a Quadratic · Substitution skeleton
  $$u = \sqrt{x}\ (\ge 0)\ \text{ or }\ u = x^2\ (\ge 0)\ \Rightarrow\ \text{quadratic in } u$$

Watch out for (7)

• The formula needs standard form first
  → Three Ways to Solve a Quadratic
• "Real roots" includes the equal case
  → The Discriminant — Nature of the Roots
• Know all three tests cold
  → Equal Roots Force a Coefficient Progression
• Difference uses (sum)² − 4·product, not (sum)² − product
  → Difference and Ratio of the Roots
• Check the sum before reaching for the formula
  → The a + b + c = 0 Shortcut
• All three conditions are needed — not just the endpoints
  → Location of the Roots in an Interval
• Reject substitution values that violate the domain
  → Equations That Reduce to a Quadratic

Formulas (7)

• Vieta's Relations — Sum and Product of Roots · Vieta's relations
  $$\alpha + \beta = -\dfrac{b}{a}, \qquad \alpha\beta = \dfrac{c}{a}$$
• Symmetric Functions & Forming New Equations · Build the equation from new sum & product
  $$x^2 - S x + P = 0, \quad S = \text{(new sum)},\ P = \text{(new product)}$$
• Means of the Roots & Equal-Magnitude Conditions · Means of the roots
  $$\text{AM} = -\tfrac{b}{2a},\quad \text{GM} = \sqrt{\tfrac{c}{a}},\quad \text{HM} = -\tfrac{2c}{b}$$
• Cross-Equation and Shared-Ratio Conditions · Subtract the substituted equations
  $$n^2+pn+m = 0,\ \ m^2+pm+n = 0 \ \Rightarrow\ (n-m)(n+m+p-1) = 0$$
• Reducing a Symmetric Equation by Substitution · Shift to the centre of symmetry
  $$(x-a)^4 + (x-b)^4 = k,\ \ u = x - \tfrac{a+b}{2} \ \Rightarrow\ u^4 + Au^2 + B = 0$$
• Self-Referential Root Conditions · Translate every condition into s and p
  $$s = \alpha+\beta,\quad p = \alpha\beta \ \Rightarrow\ \text{solve the system in } s, p$$
• Structural and Counting Root Problems · Unchanged under squaring the roots
  $$\{\alpha^2, \beta^2\} = \{\alpha, \beta\} \ \Rightarrow\ \alpha, \beta \in \{0, 1, \omega, \omega^2\}$$

Watch out for (6)

• Difference of roots uses −4p, sum of squares uses −2p
  → Symmetric Functions & Forming New Equations
• Equal magnitude opposite sign needs TWO conditions
  → Means of the Roots & Equal-Magnitude Conditions
• Subtract, don't add
  → Cross-Equation and Shared-Ratio Conditions
• "Number of real roots" ≠ "sum of all roots"
  → Reducing a Symmetric Equation by Substitution
• Don't divide away a root you still need
  → Self-Referential Root Conditions
• Enumerate the set-equality cases
  → Structural and Counting Root Problems

Formulas (6)

• Cube Roots of Unity — the x² + x + 1 Hook · The two defining facts
  $$\omega^3 = 1, \qquad 1 + \omega + \omega^2 = 0$$
• Constructed Symmetric-Coefficient Equations · Symmetric construction ⇒ unit root
  $$(q-r)x^2 + (r-p)x + (p-q) = 0 \ \Rightarrow\ x = 1,\ \ x = \tfrac{p-q}{q-r}$$
• Modulus Equations Reducing to Quadratics · Substitute t = |·| ≥ 0
  $$|x-a|^2 + |x-a| - 2 = 0,\ \ t = |x-a| \ge 0 \ \Rightarrow\ t^2 + t - 2 = 0$$
• Parametric Quadratics — Factor, Don't Force · Vertex (minimum) value, a > 0
  $$\min(ax^2+bx+c) = c - \dfrac{b^2}{4a} = -\dfrac{D}{4a}$$
• Logarithmic Equations That Are Quadratics · Name the log, solve, invert
  $$t = \log_b u \ \Rightarrow\ \text{quadratic in } t,\quad u = b^{\,t}$$
• Quadratics Built From Their Roots · Expand, then Vieta
  $$x^2 - ax - bx + (ab - c) = 0 \ \Rightarrow\ \alpha+\beta = a+b,\ \ \alpha\beta = ab - c$$

Watch out for (6)

• Reduce the exponent mod 3 before anything else
  → Cube Roots of Unity — the x² + x + 1 Hook
• Repeated root is −B/2A, not −B/A
  → Constructed Symmetric-Coefficient Equations
• A negative value of the modulus variable is impossible
  → Modulus Equations Reducing to Quadratics
• Look for the unit-root before the formula
  → Parametric Quadratics — Factor, Don't Force
• Solve for the log first, the variable second
  → Logarithmic Equations That Are Quadratics
• Expand the constructed form before reading coefficients
  → Quadratics Built From Their Roots