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Quadratic Equations formulas

21 formulas and 19 common traps for NDA Mathematics Quadratic Equations, grouped by subtopic.

Full notes with worked examples

Nature of Roots & Boundary Conditions

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What a Quadratic Equation Is

Standard form

ax2+bx+c=0,a≠0ax^2 + bx + c = 0, \quad a \neq 0

Three Ways to Solve a Quadratic

Quadratic formula

x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The Discriminant — Nature of the Roots

Discriminant

D=b2−4acD = b^2 - 4ac

Equal Roots Force a Coefficient Progression

Progression tests

GP: b2=acAP: 2b=a+cHP: 2b=1a+1c\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

∣α−β∣=(α+β)2−4αβ=D∣a∣|\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  ⟺  x=1 is a root,other root=caa + 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≥0,f(p)>0,f(q)>0,p<−b2a<qD \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=x (≥0)  or  u=x2 (≥0) ⇒ quadratic in uu = \sqrt{x}\ (\ge 0)\ \text{ or }\ u = x^2\ (\ge 0)\ \Rightarrow\ \text{quadratic in } u

Common traps

The formula needs standard form first

bb and cc in x=−b±b2−4ac2ax = \frac{-b\pm\sqrt{b^2-4ac}}{2a} are the coefficients AFTER moving everything to one side and with a>0a > 0 if you like. Reading b,cb, c off an un-rearranged equation (e.g. x2=5x−6x^2 = 5x - 6) is the most common slip.

"Real roots" includes the equal case

"Has real roots" means D≥0D \geq 0 (real and distinct OR equal). "Distinct real roots" is the strict D>0D > 0. Watch which the question asks — boundary values of a parameter live exactly at D=0D = 0.

Know all three tests cold

The HP condition 2b=1a+1c\frac{2}{b}=\frac{1}{a}+\frac{1}{c} is the one most often produced, but the wrong-progression option (AP or GP) is always offered. After simplifying D=0D=0, match the EXACT relation, don't pattern-match on "it has fractions so it's HP".

Difference uses (sum)² − 4·product, not (sum)² − product

(α−β)2=(α+β)2−4αβ(\alpha-\beta)^2 = (\alpha+\beta)^2 - 4\alpha\beta — the coefficient is 44, the same 44 as in the discriminant. Using 22 instead is the standard error.

Check the sum before reaching for the formula

Whenever the coefficients are built from symbols like (b−c),(c−a),(a−b)(b-c), (c-a), (a-b) or (q−r),(r−p),(p−q)(q-r),(r-p),(p-q), test a+b+ca+b+c first — it is almost always engineered to vanish, making x=1x=1 a free root.

All three conditions are needed — not just the endpoints

f(p)>0f(p)>0 and f(q)>0f(q)>0 alone allow BOTH roots to sit on the same side of the interval (or to be complex). You also need D≥0D \ge 0 and the vertex inside (p,q)(p,q). Drop any one and a wrong-count option catches you.

Reject substitution values that violate the domain

With u=xu = \sqrt{x} or u=x2u = x^2, a negative uu is impossible — discard it. And for a modulus branch, a candidate root is only valid if it lies in the interval that defined that branch. Forgetting this manufactures phantom roots.

Vieta's Relations & Root-Coefficient Identities

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Vieta's Relations — Sum and Product of Roots

Vieta's relations

α+β=−ba,αβ=ca\alpha + \beta = -\dfrac{b}{a}, \qquad \alpha\beta = \dfrac{c}{a}

Symmetric Functions & Forming New Equations

Build the equation from new sum & product

x2−Sx+P=0,S=(new sum), P=(new 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

AM=−b2a,GM=ca,HM=−2cb\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

n2+pn+m=0,  m2+pm+n=0 ⇒ (n−m)(n+m+p−1)=0n^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−a+b2 ⇒ u4+Au2+B=0(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=α+β,p=αβ ⇒ solve the system in s,ps = \alpha+\beta,\quad p = \alpha\beta \ \Rightarrow\ \text{solve the system in } s, p

Structural and Counting Root Problems

Unchanged under squaring the roots

{α2,β2}={α,β} ⇒ α,β∈{0,1,ω,ω2}\{\alpha^2, \beta^2\} = \{\alpha, \beta\} \ \Rightarrow\ \alpha, \beta \in \{0, 1, \omega, \omega^2\}

Common traps

Difference of roots uses −4p, sum of squares uses −2p

(α−β)2=s2−4p(\alpha-\beta)^2 = s^2 - 4p but α2+β2=s2−2p\alpha^2+\beta^2 = s^2 - 2p. Mixing the 22 and the 44 is the single most common Vieta error.

Equal magnitude opposite sign needs TWO conditions

b=0b = 0 alone only makes the roots negatives of each other IF they are real — you also need ca<0\frac{c}{a} < 0 for the roots to be real (and nonzero). A parameter value giving b=0b=0 but ca>0\frac{c}{a}>0 yields imaginary roots, not ±k\pm k.

Subtract, don't add

Adding the two substituted equations keeps a stubborn m2+n2m^2+n^2; subtracting produces the factor (m−n)(m-n) you can cancel. When you see a symmetric pair of "X is a root of … , Y is a root of …", subtract.

"Number of real roots" ≠ "sum of all roots"

After u2=−7u^2 = -7 is rejected for real roots, those two complex roots STILL count toward the sum-of-all-roots (via Vieta on the quartic). Read whether the question wants the real-root count or the full Vieta sum.

Don't divide away a root you still need

Cancelling β\beta from αβ=β\alpha\beta = \beta is valid only because β≠0\beta \neq 0 is given; the discarded case β=0\beta = 0 must be checked separately (it usually fails another condition).

Enumerate the set-equality cases

{α2,β2}={α,β}\{\alpha^2,\beta^2\}=\{\alpha,\beta\} splits into the matched case (α2=α,β2=β\alpha^2=\alpha,\beta^2=\beta) AND the swapped case (α2=β,β2=α\alpha^2=\beta,\beta^2=\alpha). The swapped case is where ω,ω2\omega, \omega^2 enter — miss it and you undercount.

Special Quadratics — Parametric, Logarithmic & Constructed

Learn this subtopic in the notes

Cube Roots of Unity — the x² + x + 1 Hook

The two defining facts

ω3=1,1+ω+ω2=0\omega^3 = 1, \qquad 1 + \omega + \omega^2 = 0

Constructed Symmetric-Coefficient Equations

Symmetric construction ⇒ unit root

(q−r)x2+(r−p)x+(p−q)=0 ⇒ x=1,  x=p−qq−r(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∣≥0 ⇒ t2+t−2=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⁡(ax2+bx+c)=c−b24a=−D4a\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⁡bu ⇒ quadratic in t,u=b tt = \log_b u \ \Rightarrow\ \text{quadratic in } t,\quad u = b^{\,t}

Quadratics Built From Their Roots

Expand, then Vieta

x2−ax−bx+(ab−c)=0 ⇒ α+β=a+b,  αβ=ab−cx^2 - ax - bx + (ab - c) = 0 \ \Rightarrow\ \alpha+\beta = a+b,\ \ \alpha\beta = ab - c

Common traps

Reduce the exponent mod 3 before anything else

ω200\omega^{200} is not a monster — 200=3(66)+2200 = 3(66) + 2, so ω200=ω2\omega^{200} = \omega^2. Always replace ωn\omega^n by ωn mod 3\omega^{n \bmod 3} first, and look for the 1+x+x21+x+x^2 factor to send a sum to zero.

Repeated root is −B/2A, not −B/A

For equal roots the single root equals the vertex x=−B2Ax = -\frac{B}{2A} (half of the sum of roots, since both roots coincide). Using −B/A-B/A (the full sum) doubles it — the offered wrong answer.

A negative value of the modulus variable is impossible

After t=∣x−a∣t = |x-a|, discard any negative tt before solving back. And for ∣f∣=g|f| = g, a candidate root is valid only if g≥0g \ge 0 there — skip this and you import roots the equation never had.

Look for the unit-root before the formula

Parametric quadratics in NDA are nearly always built to factor (often with x=1x=1 a root). Reaching for −b±D2a\frac{-b\pm\sqrt{D}}{2a} with symbolic a,b,ca,b,c is slow and error-prone — test the coefficient sum first.

Solve for the log first, the variable second

The quadratic is in t=log⁡xt = \log x, not in xx. Its roots are values of the LOG; you still have to exponentiate to recover xx. And every recovered xx must keep the original log arguments positive.

Expand the constructed form before reading coefficients

x2−ax−bx+ab−cx^2 - ax - bx + ab - c looks like it has linear coefficient −a-a or −b-b; it doesn't until you combine them into −(a+b)-(a+b). Always collect like terms to standard form before applying Vieta.

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