Quadratic Equations — vidhyapath 2027

Concept Core

Everything you need to know — zero fluff, exam-precision focus.

The Standard Form

A quadratic equation is any equation of the form ax² + bx + c = 0 where a, b, c ∈ ℝ and a ≠ 0 (if a = 0, it becomes linear — not quadratic).

ax² + bx + c = 0 (a ≠ 0)

It is a degree-2 polynomial equation, so it has exactly 2 roots (real or complex, counting multiplicity).

The Discriminant — Heart of the Chapter

The discriminant D (also written Δ) tells you the nature of roots before solving:

D = b² − 4ac

Condition	Nature of Roots
`D > 0`	2 real, distinct roots
`D = 0`	2 real, equal roots
`D < 0`	2 complex conjugate roots

Vieta's Formulas — Sum & Product

If α and β are the roots of ax² + bx + c = 0, then:

α + β = −b/a (Sum of roots)

αβ = c/a (Product of roots)

Memory trick: Sum = negative of middle / leading. Product = last / leading.

The Quadratic Formula

The universal solution — works for any quadratic:

x = (−b ± √(b² − 4ac)) / 2a

The ± gives both roots. When D = 0, both roots are identical: x = −b/2a.

Forming a Quadratic from Roots

If you know the roots α and β, construct the equation using:

x² − (α+β)x + αβ = 0

This is the reverse of Vieta's. EAPCET frequently asks: "Form the equation whose roots are..."

Symmetric Functions of Roots

Express higher powers in terms of sum and product (no need to find roots):

α² + β² = (α+β)² − 2αβ

α³ + β³ = (α+β)³ − 3αβ(α+β)

|α − β| = √D / |a|

Special Conditions — EAPCET Favourites

Roots equal in magnitude, opposite in sign → b = 0 (coefficient of x is zero)

Roots are reciprocals → a = c (leading and constant coefficients equal)

One root is zero → c = 0 (constant term is zero)

One root is double the other (α, 2α) → 2b² = 9ac

Roots differ by k → D = k²a² i.e., (α−β)² = k²

Formula Vault

Every formula you'll ever need — exam-ready, nothing skipped.

Standard Form

ax² + bx + c = 0

a, b, c ∈ ℝ, a ≠ 0

Discriminant

D = b² − 4ac

D > 0: real & distinct; D = 0: equal; D < 0: complex

Quadratic Formula

x = (−b ± √D) / 2a

Both roots in one formula

Sum of Roots

α + β = −b/a

Negative of (b/a)

Product of Roots

αβ = c/a

Direct ratio c/a

Form Equation from Roots

x² − (S)x + P = 0

S = sum, P = product of roots

Sum of Squares of Roots

α² + β² = (α+β)² − 2αβ

= (b/a)² − 2(c/a)

Sum of Cubes of Roots

α³+β³ = (α+β)³−3αβ(α+β)

Use (a+b)³ identity

Difference of Roots

|α − β| = √D / |a|

Only real when D ≥ 0

Sum of Reciprocals

1/α + 1/β = (α+β)/(αβ) = −b/c

Divide sum by product

Product of Reciprocals

1/(αβ) = a/c

Inverse of product of roots

Equal Roots Condition

b² = 4ac (D = 0)

Common root: x = −b/2a

Worked Examples

5 carefully chosen problems — from basic to EAPCET-level to trap questions. Click any to expand.

Easy Find the nature of roots of 2x² − 5x + 3 = 0 ▾

Find the nature of roots and solve: 2x² − 5x + 3 = 0

1

Identify: a = 2, b = −5, c = 3

2

Compute discriminant: D = (−5)² − 4(2)(3) = 25 − 24 = 1

3

Since D = 1 > 0, roots are real and distinct.

4

Using the formula: x = (5 ± √1) / 4 = (5 ± 1) / 4

✓ x = 6/4 = 3/2 or x = 4/4 = 1

Medium If α, β are roots of x² − 3x + 2 = 0, find α² + β² ▾

The roots of x² − 3x + 2 = 0 are α and β. Find the value of α² + β² without finding roots.

1

From Vieta's: α + β = 3 (= −(−3)/1) and αβ = 2 (= 2/1)

2

Apply the identity: α² + β² = (α+β)² − 2αβ

3

Substitute: = (3)² − 2(2) = 9 − 4 = 5

✓ α² + β² = 5

Medium Find k such that roots of kx² + 4x + 1 = 0 are equal ▾

Find the value of k for which the equation kx² + 4x + 1 = 0 has equal roots.

1

For equal roots: D = 0. Here a = k, b = 4, c = 1.

2

Set D = b² − 4ac = 0: so (4)² − 4(k)(1) = 0

3

16 − 4k = 0 → 4k = 16 → k = 4

4

Check: k ≠ 0 (so it remains quadratic). ✓

✓ k = 4

EAPCET Level If α, β are roots of x² − px + q = 0, form equation with roots α² + β², αβ ▾

α and β are roots of x² − px + q = 0. Form a quadratic equation with roots α² + β² and αβ.

1

From given equation: α + β = p, αβ = q

2

New roots are r₁ = α² + β² and r₂ = αβ = q

3

Calculate r₁: α² + β² = (α+β)² − 2αβ = p² − 2q

4

Sum of new roots: S = (p² − 2q) + q = p² − q

5

Product of new roots: P = (p² − 2q)(q) = q(p² − 2q)

6

New equation: x² − Sx + P = 0

✓ x² − (p² − q)x + q(p² − 2q) = 0

Trap Question Find the sum of reciprocals of roots of 3x² − 5x + 7 = 0 ▾

If α, β are roots of 3x² − 5x + 7 = 0, find 1/α + 1/β. ⚠️ Most students get this wrong — don't fall for it.

1

The common trap: students try to find roots using the formula and then take reciprocals — messy and error-prone with complex numbers.

2

The smart way: 1/α + 1/β = (α + β) / (αβ) — just divide sum by product.

3

From Vieta's: α + β = 5/3 and αβ = 7/3

4

So: 1/α + 1/β = (5/3) ÷ (7/3) = (5/3) × (3/7) = 5/7

✓ 1/α + 1/β = 5/7 (Note: D < 0 so roots are complex, but answer is real!)

Mistake DNA

The 5 most common errors in EAPCET — sourced from distractor analysis of real exam questions.

⚡

Sign Error in Vieta's Sum Formula

The most frequent error. Students write sum = b/a instead of −b/a.

❌ Wrong

For x² − 5x + 6 = 0:
α + β = 5/1 = 5 ✗
(b = −5, so b/a = −5)

✓ Correct

α + β = −b/a = −(−5)/1 = +5 ✓
The formula is −b/a.
The sign of b matters.

In x² − 5x + 6, b = −5. So −b/a = −(−5)/1 = 5. The answer is still 5 but by correct reasoning. When b is positive, sum is negative.

🎯

Discriminant Computed as 4ac − b² (reversed)

Students swap the order, flipping the sign of D and misidentifying the nature of roots.

❌ Wrong

D = 4ac − b²
= 4(2)(1) − 9 = −1
Says: complex roots ✗

✓ Correct

D = b² − 4ac
= 9 − 4(2)(1) = 1
Real, distinct roots ✓

Remember the sequence in the quadratic formula: x = (−b ± √(b²−4ac))/2a. b² comes first.

🔄

Forgetting to Normalise when a ≠ 1

For 2x² − 6x + 4 = 0, students read sum = 6 instead of 6/2 = 3.

❌ Wrong

2x² − 6x + 4 = 0:
α + β = 6 ✗
αβ = 4 ✗

✓ Correct

α + β = −(−6)/2 = 3 ✓
αβ = 4/2 = 2 ✓
Always divide by a.

The formulas are α+β = −b/a and αβ = c/a. When a = 1, division doesn't change the value, so students forget it for other cases.

🧮

Trying to Find Roots When Symmetric Functions Are Enough

Wasting 3 minutes solving the equation when Vieta's gives the answer in 10 seconds.

❌ Wrong Approach

Solve x² − 7x + 12 = 0 to get
α = 3, β = 4, then compute
α² + β² = 9 + 16 = 25 ✗ (slow)

✓ Smart Way

α+β = 7, αβ = 12
α²+β² = 49 − 24 = 25 ✓
No roots needed. Faster!

EAPCET is time-pressured. If a question involves α² + β², α³ + β³, 1/α + 1/β — always use Vieta's identities. Finding roots first is a trap.

🌀

Claiming Complex Roots "Don't Exist" or Are Invalid

When D < 0, some students write "no solution" or skip the question.

❌ Wrong

"D = −3 < 0, so no roots"
Or: "undefined" ✗

✓ Correct

D < 0: roots are complex
conjugates: α = p + qi,
β = p − qi. Both valid. ✓

In EAPCET, "no real roots" is a valid and acceptable answer. Complex conjugate roots always come in pairs when coefficients are real.

Chapter Intelligence

Data-driven exam insights. Know exactly what to expect and where to spend your time.

EAPCET Topic Weightage (2019–2024)

Nature of roots (Discriminant)

~9

Vieta's formulas (sum/product)

~7

Form equation from roots

~5

Symmetric functions (α²+β²)

~5

Condition for special roots

~4

Quadratic formula

~2

High-Yield PYQ Patterns

If α, β are roots, find α² + β² Find k for equal roots Form equation with new roots Sum of 1/α + 1/β Roots α, 2α — find relation Product condition for real roots When do roots have same sign?

Exam Strategy — Score Maximally in 90 Seconds per Question

If the question mentions roots α and β — immediately write down α+β and αβ using Vieta's. Do this before reading the full question. It primes your brain.
For "nature of roots" questions, compute D first. If D is a perfect square, the roots are also rational. This narrows your answer choices instantly.
When forming a new equation from modified roots (like α+2, β+2), compute the new sum and new product — never try to find the original roots first.
Questions with "for all real x" or "minimum value" are disguised quadratic questions — use the vertex formula x = −b/2a for the critical point.
If you see 1/α + 1/β, it equals (α+β)/(αβ) = −b/c. Direct. No roots needed.
For equal roots questions: always check that the resulting value keeps a ≠ 0, or you'll lose the mark on a technicality.

Connected Chapters

Complex Numbers — when D < 0, roots are complex. Study together.
Inequalities — sign of quadratic expression depends on D and a.
Matrices — characteristic equations are quadratic polynomials.
Coordinate Geometry — intersection of curves leads to quadratics.

Time Budget in Exam

Discriminant/nature questions — 45 seconds max. Pure substitution.
Vieta's/symmetric function — 60–75 seconds. Standard.
Form equation from roots — 90 seconds. Two-step.
Complex manipulation (α³+β³) — 90–120 seconds. Use identity chain.