Quadratic Equations (Class 10): Notes और MCQ Practice

आसान भाषा में

Quadratic Equation वह algebraic equation है जिसमें variable की highest power (degree) 2 होती है। इसका standard form ax² + bx + c = 0 होता है (जहाँ a ≠ 0)। किसी भी quadratic equation के maximum 2 solutions यानी roots होते हैं।

Important Points & Formulas

• Standard Form: ax² + bx + c = 0 (जहाँ a, b, c real numbers हैं और a ≠ 0).
• Quadratic Formula: x = (-b ± √(b² - 4ac)) ÷ 2a
• Discriminant (D): D = b² - 4ac
• Nature of Roots:
• यदि D > 0: Two distinct real roots (दो अलग-अलग वास्तविक मूल)
• यदि D = 0: Two equal real roots (दो बराबर मूल, x = -b ÷ 2a)
• यदि D < 0: No real roots (कोई वास्तविक मूल नहीं)
• Solving Methods: Factorisation (splitting the middle term) और Quadratic Formula method.

2 Memory Tricks

1. D decides Destiny: Discriminant (D) ही roots की किस्मत तय करता है—Positive है तो 2 अलग roots, Zero है तो Equal twins (बराबर roots), Negative है तो No real roots!
2. Opposite 'b': Formula याद रखें "-b means opposite of b"—अगर equation में +5x है तो formula में -5 लिखें, और अगर -5x है तो +5 लिखें।

2 Common Board-Exam Mistakes

1. Whole Division भूलना: Formula लगाते वक्त बच्चे अक्सर सिर्फ √D के नीचे 2a लिख देते हैं। याद रखें: पूरा (-b ± √D) 2a से divide होता है।
2. -4ac में Signs की गड़बड़ी: अगर 'c' negative हो, तो formula का minus और c का minus मिलकर plus हो जाता है (-4 × a × -c = +4ac)। यहाँ sign check करना न भूलें!

Class 10 Mathematics: Quadratic Equations

1. Chapter in Short

A quadratic equation is an equation of degree 2, meaning the highest power of the variable (usually $x$) is 2. In this chapter, we learn:

• How to identify quadratic equations.
• How to write them in standard form: $ax^2 + bx + c = 0$.
• Methods to find their solutions (called roots): Factorisation and the Quadratic Formula.
• How to check the nature of roots using the Discriminant ($D$) without solving the entire equation.

2. Key Concepts Explained Simply

What is a Quadratic Equation?

Imagine you want to design a rectangular prayer hall. You are told that the length of the hall is 2 metres more than its breadth, and its total floor area is 35 square metres.

• Let the breadth be $x$ metres.
• Then the length is $(x + 2)$ metres.
• Area = Length × Breadth = $(x + 2) × x = x^2 + 2x$.
• Since Area = 35, we write: $x^2 + 2x = 35$, or $x^2 + 2x - 35 = 0$.

Because the highest power of $x$ is 2, this is a quadratic equation.

What is a "Root" of an Equation?

A root (or solution) is simply a real number which, when substituted in place of $x$, makes the left-hand side equal to zero. A quadratic equation can have at most two real roots.

Methods to Solve a Quadratic Equation

1. Factorisation (Splitting the Middle Term):

• Write $ax^2 + bx + c = 0$.
• Split the middle term $bx$ into two terms such that their sum equals $b$ and their product equals $a × c$.
• Take out common terms to make two linear factors.

1. The Quadratic Formula (Sridharacharya's Rule):

• A universal formula that directly gives the answers, especially useful when splitting the middle term is hard or impossible.
• $x = [-b ± \sqrt{b^2 - 4ac}] ÷ (2a)$

Nature of Roots

Before solving, we look at the part under the square root: $b^2 - 4ac$. This value is called the Discriminant ($D$).

• If $D > 0$: The equation has two distinct (different) real roots.
• If $D = 0$: The equation has two equal real roots (both roots are $-b ÷ (2a)$).
• If $D < 0$: The equation has no real roots (because we cannot find the square root of a negative real number).

3. Important Definitions, Formulas, and Facts

• Standard Form: $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a ≠ 0$.
• Discriminant Formula: $D = b^2 - 4ac$
• Quadratic Formula:

$x = [-b ± \sqrt{D}] ÷ (2a) = [-b ± \sqrt{b^2 - 4ac}] ÷ (2a)$

• Conditions for Roots:
• $b^2 - 4ac > 0 \implies$ Two distinct real roots
• $b^2 - 4ac = 0 \implies$ Two equal (coincident) real roots
• $b^2 - 4ac < 0 \implies$ No real roots
• Zero-Product Rule: If $A × B = 0$, then either $A = 0$ or $B = 0$ (or both).
• Historical Fact: Ancient Indian mathematicians like Brahmagupta (around 628 CE) and Sridharacharya (around 1025 CE) gave explicit rules and formulas to solve quadratic equations.

4. Solved Board-Style Examples

Example 1: Check whether an equation is quadratic

Question: Check whether $(x - 2)^2 + 1 = 2x - 3$ is a quadratic equation.

Solution:

1. Expand the Left-Hand Side (LHS):

$(x - 2)^2 + 1 = (x^2 - 4x + 4) + 1 = x^2 - 4x + 5$

1. Set LHS equal to the Right-Hand Side (RHS):

$x^2 - 4x + 5 = 2x - 3$

1. Move all terms to the LHS:

$x^2 - 4x - 2x + 5 + 3 = 0$

$x^2 - 6x + 8 = 0$

1. Since this is in the form $ax^2 + bx + c = 0$ with $a = 1 ≠ 0$, yes, it is a quadratic equation.

Example 2: Solving by Factorisation

Question: Find the roots of the equation $2x^2 - 5x + 3 = 0$.

Solution:

1. Here, $a = 2, b = -5, c = 3$.
2. Product: $a × c = 2 × 3 = 6$. Sum: $b = -5$.
3. Find two numbers whose product is $6$ and sum is $-5$. These numbers are $-2$ and $-3$.
4. Split the middle term:

$2x^2 - 2x - 3x + 3 = 0$

1. Group terms:

$2x(x - 1) - 3(x - 1) = 0$

$(2x - 3)(x - 1) = 0$

1. Set each factor to zero:

• $2x - 3 = 0 \implies 2x = 3 \implies x = 3 ÷ 2$
• $x - 1 = 0 \implies x = 1$

Answer: The roots are $x = 1$ and $x = 3 ÷ 2$.

Example 3: Solving using the Quadratic Formula

Question: Solve $2x^2 - 7x + 3 = 0$ using the quadratic formula.

Solution:

1. Compare with $ax^2 + bx + c = 0$:

$a = 2, b = -7, c = 3$

1. Find the discriminant $D$:

$D = b^2 - 4ac = (-7)^2 - 4 × 2 × 3 = 49 - 24 = 25$

1. Since $D > 0$, there are two distinct real roots.
2. Apply the formula:

$x = [-b ± \sqrt{D}] ÷ (2a)$

$x = [-(-7) ± \sqrt{25}] ÷ (2 × 2)$

$x = [7 ± 5] ÷ 4$

1. Calculate both values:

• $x = (7 + 5) ÷ 4 = 12 ÷ 4 = 3$
• $x = (7 - 5) ÷ 4 = 2 ÷ 4 = 1 ÷ 2$

Answer: The roots are $x = 3$ and $x = 1 ÷ 2$.

Example 4: Condition for Equal Roots (Very Important for Boards)

Question: Find the value of $k$ for which the quadratic equation $2x^2 + kx + 3 = 0$ has two equal real roots.

Solution:

1. Compare with $ax^2 + bx + c = 0$:

$a = 2, b = k, c = 3$

1. For equal roots, the discriminant must be zero ($D = 0$):

$b^2 - 4ac = 0$

1. Substitute the values:

$k^2 - 4 × 2 × 3 = 0$

$k^2 - 24 = 0$

$k^2 = 24$

1. Take square root on both sides:

$k = ±\sqrt{24} = ±\sqrt{4 × 6} = ±2\sqrt{6}$

Answer: $k = 2\sqrt{6}$ or $k = -2\sqrt{6}$.

Example 5: Real-Life Word Problem

Question: Find two numbers whose sum is 27 and product is 182.

Solution:

1. Let the first number be $x$.
2. Since their sum is 27, the second number is $(27 - x)$.
3. Their product is 182:

$x × (27 - x) = 182$

$27x - x^2 = 182$

1. Rearrange to standard form:

$x^2 - 27x + 182 = 0$

1. Factorise: Find two numbers whose product is 182 and sum is $-27$. The numbers are $-13$ and $-14$.

$x^2 - 13x - 14x + 182 = 0$

$x(x - 13) - 14(x - 13) = 0$

$(x - 13)(x - 14) = 0$

1. Therefore, $x = 13$ or $x = 14$.

Answer: The two required numbers are 13 and 14.

5. Common Mistakes and Quick Revision Points

Common Mistakes to Avoid:

• Sign Error with $-b$: If $b = -5$, then $-b = -(-5) = +5$. Many students write $-5$.
• Sign Error in $b^2$: $(-4)^2 = +16$, NOT $-16$. A squared real number is always non-negative.
• Dividing by $2$ instead of $2a$: In the quadratic formula, the denominator is $2a$. If $a = 3$, the denominator is $2 × 3 = 6$, not just $2$.
• Not rearranging first: Always convert an equation into standard form $ax^2 + bx + c = 0$ before finding $a, b, c$. For example, in $3x^2 = 5x - 2$, first write $3x^2 - 5x + 2 = 0$.
• Forgetting the $±$ sign: When solving $k^2 = 24$, do not write just $k = \sqrt{24}$; remember that $k = ±\sqrt{24}$.

Quick Revision Checklist:

• Degree of quadratic equation = 2.
• Standard form: $ax^2 + bx + c = 0$ with $a ≠ 0$.
• Maximum number of roots = 2.
• $D > 0 \implies$ 2 real and distinct roots.
• $D = 0 \implies$ 2 real and equal roots.
• $D < 0 \implies$ No real roots.