Quadratic Equations

Welcome to your ultimate and comprehensive study guide for Chapter 1: Quadratic Equations. In mathematics, equations form the fundamental basis for solving complex real-world problems. This chapter covers the definition, standard forms, and the three primary methods used to find the roots of second-degree equations. Whether you are preparing for your board exams or building a strong mathematical foundation, this guide contains all the definitions, step-by-step examples, and conceptual insights you need.

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1. What is a Quadratic Equation?

An equation which contains the square of the unknown (variable) quantity, but no higher power, is called a quadratic equation or an equation of the second degree.

• Standard Form: The general or standard form of a quadratic equation in one variable x is written as:
  ax² + bx + c = 0
  (Where a ≠ 0, and a, b, c are real numbers). Here, “a” is the coefficient of x², “b” is the coefficient of x, and “c” is the constant term.
• Pure Quadratic Equation: If the coefficient of x is zero (i.e., b = 0) in a standard quadratic equation, it reduces to ax² + c = 0. This is called a pure quadratic equation. For example: x² – 16 = 0.

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2. Methods to Solve Quadratic Equations

To find the solution set (roots) of a quadratic equation, mathematicians primarily use three reliable methods:

1. Factorization: Breaking the equation into linear factors.
2. Completing the Square: Manipulating the equation to form a perfect square.
3. Quadratic Formula: Using a derived mathematical formula for direct calculation.

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3. Solution by Factorization

In this method, we write the quadratic equation in the standard form (ax² + bx + c = 0) and split the middle term (bx) into two parts such that their sum equals “b” and their product equals “ac”.

Example: Solve 3x² – 6x = x + 20 by factorization.
Step-by-Step Solution:

• Step 1: Write in standard form: Bring all terms to one side.
  3x² – 6x – x – 20 = 0 → 3x² – 7x – 20 = 0
• Step 2: Find suitable factors: We need two numbers that add up to -7 and multiply to (3 × -20) = -60. These numbers are -12 and +5.
• Step 3: Split the middle term:
  3x² – 12x + 5x – 20 = 0
• Step 4: Factorize:
  3x(x – 4) + 5(x – 4) = 0
  (x – 4)(3x + 5) = 0
• Step 5: Find the roots:
  Either (x – 4) = 0 → x = 4
  Or (3x + 5) = 0 → 3x = -5 → x = -5/3
• Solution Set: {-5/3, 4}

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4. Solution by Completing the Square

Sometimes, an equation cannot be easily factorized. In such cases, completing the square is highly effective.

Example: Solve x² – 3x – 4 = 0
Step-by-Step Solution:

• Step 1: Shift the constant term: x² – 3x = 4
• Step 2: Add the square of half of the x-coefficient to both sides: The coefficient of x is -3. Half of it is -3/2. Square it and add to both sides.
  x² – 3x + (-3/2)² = 4 + (-3/2)²
• Step 3: Complete the square:
  (x – 3/2)² = 4 + 9/4
  (x – 3/2)² = 25/4
• Step 4: Take the square root of both sides:
  x – 3/2 = ±5/2
  x = 3/2 ± 5/2
• Result: x = 4 or x = -1. Solution Set: {-1, 4}

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5. The Quadratic Formula

The quadratic formula is derived from the standard form (ax² + bx + c = 0) using the method of completing the square. It is a universal tool that can solve any quadratic equation, even when factorization fails.

The Formula:
x = [-b ± √(b² – 4ac)] / 2a

How to use it: Simply identify the values of a, b, and c from your standard equation and plug them into the formula to find the roots (x).

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6. Equations Reducible to Quadratic Form

In higher-level problems, you will encounter equations that do not look quadratic initially but can be transformed into the standard quadratic form using appropriate substitutions. These include:

• Biquadratic Equations: Equations of the type ax⁴ + bx² + c = 0. (Substitute x² = y).
• Exponential Equations: Equations where the variable occurs in the exponent (e.g., 5^1+x + 5^1-x = 26).
• Reciprocal Equations: Equations that remain unchanged when x is replaced by 1/x.
• Radical Equations: Equations involving an expression under a radical sign (√). Remember to always check for extraneous roots (solutions that do not satisfy the original equation) when solving radical equations!

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Essential Conceptual Review Questions

Q1: What happens if ‘a’ becomes zero in the standard quadratic equation (ax² + bx + c = 0)?
Answer: If a = 0, the x² term disappears, and the equation reduces to a linear equation (bx + c = 0). This is why the definition strictly states that “a” cannot be equal to zero.

Q2: Why must we check our answers when solving radical equations?
Answer: When solving radical equations, we often square both sides to eliminate the radical sign. Squaring can introduce additional solutions that are mathematically correct for the squared equation but do not satisfy the original radical equation. These are called extraneous roots and must be discarded.