Vardaan Learning Institute - Class 10 Quadratic Equations Classroom

CONCEPT 01 / 05 TOPIC 1 CLASS 10 NOTES

TOPIC 1: STANDARD FORM & PARABOLA ORIENTATION

📌 Standard Form Definition

\( ax^2 + bx + c = 0 \quad \text{where } a \neq 0, \text{ and } a, b, c \in \mathbb{R} \)

Polynomial equation of degree 2. Leading coefficient \( a \) cannot be zero (\( a \neq 0 \)).

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CONCEPT 02 / 05 TOPIC 2 CLASS 10 NOTES

TOPIC 2: METHODS OF SOLVING & X-INTERCEPT ROOTS

1️⃣ Middle-Term Splitting

Split \( bx = px + qx \) where \( p+q=b \) and \( p \times q = a \times c \).

2️⃣ Quadratic Formula

\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

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CONCEPT 03 / 05 TOPIC 3 CLASS 10 NOTES

TOPIC 3: DISCRIMINANT \( D = b^2 - 4ac \) & NATURE OF ROOTS

\( D > 0 \)

2 Real & Distinct Roots

\( D = 0 \)

2 Equal Real Roots

\( D < 0 \)

No Real Roots

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CONCEPT 04 / 05 TOPIC 4 CLASS 10 NOTES

TOPIC 4: VIETA'S FORMULAS & VERTEX SYMMETRY

💡 Vieta's Relations

\( \alpha + \beta = -\frac{b}{a}, \quad \alpha \cdot \beta = \frac{c}{a} \)

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CONCEPT 05 / 05 TOPIC 5 CLASS 10 NOTES

TOPIC 5: REAL-WORLD PARABOLIC TRAJECTORY

⚽ Football Flight Trajectory

Height \( h(t) = -5t^2 + 20t \). Max Height = 20 m at \( t = 2 \text{ s} \). Air time = 4 s.

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QUESTION 01Standard Form & IdentificationClass 10 Core

Check whether \( (x + 1)^2 = 2(x - 3) \) is a quadratic equation. If yes, express it in standard form \( ax^2 + bx + c = 0 \) and state the values of \( a, b, c \).

Given Equation: \( (x + 1)^2 = 2(x - 3) \)

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Detailed Solution 01

Step 1: Expand LHS:
\( x^2 + 2x + 1 = 2x - 6 \)

Step 2: Rearrange to RHS = 0:
\( x^2 + 2x - 2x + 1 + 6 = 0 \implies x^2 + 7 = 0 \)

Conclusion: Yes, it is a quadratic equation of degree 2 with \( a = 1, b = 0, c = 7 \).

QUESTION 02Factorisation MethodClass 10 Standard

Solve the quadratic equation \( 2x^2 - 5x + 3 = 0 \) by middle-term splitting factorisation.

Given Equation: \( 2x^2 - 5x + 3 = 0 \)

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Detailed Solution 02

Splitting Middle Term: Product = \( 2 \times 3 = 6 \), Sum = \( -5 \). Factors: \( -2 \) and \( -3 \).

\( 2x^2 - 2x - 3x + 3 = 0 \)
\( 2x(x - 1) - 3(x - 1) = 0 \)
\( (2x - 3)(x - 1) = 0 \)

Roots: \( x = \frac{3}{2} \) or \( x = 1 \).