CH – 4 QUADRATIC EQUATIONS

Master degree-2 equations with simple steps and boost your board exam confidence!

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Chapter Overview

In this chapter, you will learn:

• What a quadratic equation is and how to write it.
• How to check if a given equation is quadratic.
• How to solve quadratic equations using factorisation.
• How to solve quadratic equations by completing the square.
• How to use the quadratic formula to find answers directly.
• How to find the nature of answers using the discriminant.
• How to change simple word problems into quadratic equations.

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🎯 Board Exam Focus

Here are the important skills you should practice carefully:

• Identifying quadratic equations.
• Writing equations in standard form.
• Solving by factorisation.
• Solving by completing the square.
• Solving using the quadratic formula.
• Finding the discriminant to check the nature of answers.
• Solving simple word problems.

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Most Important Concepts

What is a Quadratic Equation?

A quadratic equation is an equation where the highest power of the variable is 2.

The standard form of a quadratic equation is:

ax² + bx + c = 0

Here:

• x is the variable.
• a, b, and c are real numbers.
• a cannot be equal to 0. (If a = 0, the term containing x² becomes zero, and it is no longer quadratic).

Simple Examples:

• 2x² + x − 300 = 0
• x² − 3x + 2 = 0
• 3x² − 5 = 0 (Here b = 0, which is completely fine)

How to Identify Equations That Are NOT Quadratic:

• Power is not 2: x³ − 2x + 1 = 0 is not quadratic because the highest power is 3.
• a is zero: 0x² + 2x + 3 = 0 simplifies to 2x + 3 = 0, which is linear, not quadratic.
• Variable in the denominator or under root: x + 1/x = 2 becomes x² + 1 = 2x after solving, but in its initial form, it must be simplified first.

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Solving by Factorisation

This method is also called splitting the middle term.

Step-by-Step Method:

1. Write the equation in standard form: ax² + bx + c = 0.
2. Multiply a and c to get the product a × c.
3. Find two numbers that multiply to give a × c and add up to give b.
4. Split the middle term bx using these two numbers.
5. Group the terms into pairs and take out common factors.
6. Use the zero-product rule to find the answers.

Zero-Product Rule: If two numbers multiplied together give zero (A × B = 0), then either A = 0 or B = 0.

Simple Example:

Solve x² − 5x + 6 = 0.

1. Here, a = 1, b = −5, and c = 6.
2. Product = 1 × 6 = 6. Sum = −5.
3. The two numbers are −2 and −3 because (−2) × (−3) = 6 and (−2) + (−3) = −5.
4. Split the middle term: x² − 2x − 3x + 6 = 0.
5. Group terms: x(x − 2) − 3(x − 2) = 0.
6. Take common bracket: (x − 2)(x − 3) = 0.
7. Apply zero-product rule:
   • x − 2 = 0 gives x = 2
     
   • x − 3 = 0 gives x = 3
     

Remember: A quadratic equation can have at most two answers (called roots).

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Solving by Completing the Square

This method changes the quadratic equation into a perfect square format like (x + p)² = q.

Step-by-Step Method:

1. Make sure the coefficient of x² is 1. (If it is not 1, divide the entire equation by a).
2. Move the constant number c to the right side of the equation.
3. Look at the number in front of x (which is b).
4. Take half of b, square it, and add it to both sides of the equation.
5. Rewrite the left side as a perfect square: (x + half of b)².
6. Take the square root on both sides to find x.

Simple Example:

Solve x² + 4x − 5 = 0.

1. The number in front of x² is already 1.
2. Move −5 to the right side: x² + 4x = 5.
3. The number in front of x is 4. Half of 4 is 2.
4. Square of 2 is 2² = 4. Add 4 to both sides:x² + 4x + 4 = 5 + 4x² + 4x + 4 = 9
5. Rewrite the left side as a perfect square: (x + 2)² = 9.
6. Take square root on both sides:x + 2 = ±√9x + 2 = ±3
7. Solve for x:
   • x + 2 = 3 gives x = 1
   • x + 2 = −3 gives x = −5

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Solving Using the Quadratic Formula

When splitting the middle term is difficult, you can use the quadratic formula directly.

For the equation ax² + bx + c = 0, the formula is:

x = (−b ± √(b² − 4ac)) ÷ (2a)

Meaning of Terms:

• a is the number with x².
• b is the number with x.
• c is the constant number without any variable.

Simple Example:

Solve x² − 7x + 12 = 0.

1. Identify values: a = 1, b = −7, c = 12.
2. Find b² − 4ac:(−7)² − 4(1)(12) = 49 − 48 = 1
3. Put values into formula:x = (−(−7) ± √1) ÷ (2 × 1)x = (7 ± 1) ÷ 2
4. Calculate two answers:
   • x = (7 + 1) ÷ 2 = 8 ÷ 2 = 4
   • x = (7 − 1) ÷ 2 = 6 ÷ 2 = 3

Common Mistake: If b is negative, like b = −7, then −b becomes −(−7) = +7. Always write negative numbers inside brackets!

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🧠 The Discriminant

The value b² − 4ac inside the quadratic formula is called the discriminant. We denote it by D.

D = b² − 4ac

The value of D tells us what kind of answers (roots) we will get:

• If D > 0 (Positive number): We get two different real answers.
• If D = 0 (Zero): We get two equal real answers. Both answers are −b ÷ (2a).
• If D < 0 (Negative number): There are no real answers.

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📌 Important Formulas & Results

• Standard Form: ax² + bx + c = 0 (where a ≠ 0)
• Discriminant:D = b² − 4ac
  
• Quadratic Formula:x = (−b ± √D) ÷ (2a)
  
• Two Different Real Roots:D > 0
  
• Two Equal Real Roots: D = 0 (Answers are x = −b ÷ (2a))
• No Real Roots:D < 0
  

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🧠 How to Choose the Best Method?

• Factorisation Method: Use this when you can easily find two numbers to split the middle term. It is the fastest method for simple equations.
• Quadratic Formula: Use this when numbers are large, or when square root numbers like √2 or √3 are present, or when splitting the middle term is hard.
• Completing the Square: Use this when the question specifically asks you to solve by completing the square.

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🧠 Concepts Students Find Difficult

• Finding correct values of a, b, and c: Always rearrange the equation into ax² + bx + c = 0 first. If an equation is written as 3x + 2x² − 5 = 0, rewrite it as 2x² + 3x − 5 = 0 so a = 2, b = 3, c = −5.
• Handling positive and negative signs: Remember that (−5)² = +25, not −25. Also, −(−b) becomes positive.
• Understanding D < 0: If D comes out negative (like −7), do not calculate square root. Simply write “No real roots exist”.
• Word problems: Read the question slowly. Assume the unknown value as x. Change statements into simple mathematical relations step by step.

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⚠️ Common Mistakes Students Make

1. Forgetting that a cannot be 0:
   • Mistake: Thinking 0x² + 3x + 4 = 0 is a quadratic equation.
   • How to avoid: Check that the x² term is present and a ≠ 0.
2. Incorrect sign placement for −b:
   • Mistake: Writing −5 instead of +5 when b = −5.
   • How to avoid: Always use brackets: −b = −(−5) = 5.
3. Dividing only part of the expression by 2a:
   • Mistake: Writing x = −b ± (√D ÷ 2a).
   • How to avoid: Remember that 2a divides the entire top part: (−b ± √D) ÷ (2a).
4. Forgetting the plus-minus (±) sign:
   • Mistake: Writing x = √9 = 3 only.
   • How to avoid: Always write ± when taking square roots to solve an equation: x = ±3.
5. Not writing equations in standard form first:
   • Mistake: Solving x(x + 1) = 6 directly without multiplying.
   • How to avoid: Expand first to get x² + x − 6 = 0 before identifying a, b, c.
6. Keeping negative answers for distance or age:
   • Mistake: Giving x = −5 as the age or length in a word problem.
   • How to avoid: Reject negative values if the quantity cannot be negative in real life.

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🔥 Priority Revision

⭐⭐⭐ HIGH-PRIORITY PRACTICE

• Finding the value of k when roots are equal (D = 0).
• Solving equations using the Quadratic Formula.
• Word problems on speed, time, distance, and numbers.

⭐⭐ IMPORTANT PRACTICE

• Solving equations by Factorisation.
• Checking if a given equation is quadratic.

⭐ ADDITIONAL REVISION

• Solving equations by Completing the Square.
• Word problems based on area and perimeter.

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⚡ Quadratic Equations Quick Revision Sheet

• Standard Form: ax² + bx + c = 0 (a ≠ 0)
• Methods to Solve:
  1. Factorisation
  2. Completing the Square
  3. Quadratic Formula
• Quadratic Formula:x = (−b ± √(b² − 4ac)) ÷ (2a)
  
• Discriminant:D = b² − 4ac
  
• Nature of Roots:
  • D > 0 → Two different real answers
  • D = 0 → Two equal real answers
  • D < 0 → No real answers
• Important Reminder: Always check for sign mistakes when substituting negative numbers into formulas!

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📝 Test Yourself

1. Check whether (x − 1)² = 2(x − 3) is a quadratic equation.
2. Solve x² − 9x + 20 = 0 by factorisation.
3. Solve x² + 4x − 12 = 0 by completing the square method.
4. Solve 2x² − 5x + 3 = 0 using the quadratic formula.
5. Find the discriminant of 3x² − 2x + 1 = 0 and write the nature of its roots.
6. Find the value of k if 2x² + kx + 3 = 0 has two equal real roots.
7. Check if x = 2 is a solution of x² − 4x + 4 = 0.
8. The product of two consecutive positive integers is 156. Find the numbers.
9. The length of a rectangular field is 3 metres more than its breadth. If the area is 40 square metres, find its dimensions.
10. Two numbers have a sum of 15 and their product is 56. Find the numbers.

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🎯 Before Your Exam

☐ I can identify a quadratic equation. ☐ I know the standard form ax² + bx + c = 0. ☐ I can solve equations by factorisation. ☐ I can solve equations by completing the square. ☐ I can use the quadratic formula accurately. ☐ I know how to calculate D = b² − 4ac. ☐ I can determine the nature of roots using D. ☐ I can form and solve simple word problems.

Believe in yourself, read every question carefully, and do your absolute best!