Quadratic Equations — Lesson
1) Hook — A Fun Real-Life Story to Grab Attention
Imagine you are helping your friend design a rectangular garden where the length is 3 meters more than the width. The total area of the garden is 40 m2. How do you find the length and width? This problem can be solved using a quadratic equation, a powerful tool that helps us find unknown values in many real-life situations like this!
2) Core Concepts — Understanding Quadratic Equations
A quadratic equation in one variable x is an equation of the form:
Here, a is the coefficient of x2, b is the coefficient of x, and c is the constant term.
Example 1: Solve the quadratic equation x2 - 5x + 6 = 0.
We can factorize it:
| x2 - 5x + 6 | = (x - 2)(x - 3) |
Set each factor to zero:
- x - 2 = 0 ⇒ x = 2
- x - 3 = 0 ⇒ x = 3
So, the solutions are x = 2 and x = 3.
Example 2: Using the garden problem:
Let width = x meters, then length = x + 3 meters.
Area = length × width = 40 m2
So, x(x + 3) = 40 ⇒ x2 + 3x - 40 = 0
Factorizing:
| x2 + 3x - 40 | = (x + 8)(x - 5) |
Set each factor to zero:
- x + 8 = 0 ⇒ x = -8 (not possible as width cannot be negative)
- x - 5 = 0 ⇒ x = 5
So, width = 5 m and length = 8 m.
3) Key Formulas / Rules
Quadratic Formula:
x = -b ± √(b2 - 4ac) / 2a
Discriminant (D): D = b2 - 4ac
- If D > 0, two distinct real roots.
- If D = 0, one real root (repeated root).
- If D < 0, no real roots (roots are imaginary).
4) Did You Know?
The word “quadratic” comes from the Latin word “quadratus” meaning square. This is because the variable is squared (x2)! Quadratic equations have been studied since ancient Indian mathematicians like Bhaskara II (12th century) who gave methods to solve them.
5) Exam Tips — Common Mistakes & Board Exam Patterns
- Always check the coefficient of x2 (a ≠ 0). If a = 0, it’s not quadratic.
- Factorization: Look for two numbers whose product is ac and sum is b.
- Use quadratic formula carefully: Write the discriminant correctly and simplify under the square root.
- Sign errors: Be careful with signs when substituting values.
- Check roots: Substitute back to verify solutions.
- Board Pattern: Questions include factorization, quadratic formula, word problems (e.g., areas, motion), and discriminant interpretation.
- Mnemonic to remember quadratic formula: "x equals negative b, plus or minus the square root, of b squared minus 4 a c, all over 2 a."
Quadratic Equations — Mcq
Quadratic Equations — Mnemonic
Mnemonic 1: Quadratic Formula Song 🎵
"x equals minus b, plus minus root, b square minus 4ac, all over 2a, easy to compute!" 🎶
Tip: Sing this line twice before solving any quadratic equation to remember the formula perfectly!
Mnemonic 2: Funny Hindi Phrase for Quadratic Formula 🧠
"Minus B ko plus-minus karo, b² minus 4ac ka root nikalo, 2a se divide kar do, solution mil jaega bro!" 😎
Meaning: Just remember this casual Hindi phrase to recall the formula steps in order!
Mnemonic 3: FACT Trick for Quadratic Roots 🌟
- Find the coefficients (a, b, c)
- Apply the formula: x = [-b ± √(b² - 4ac)] / 2a
- Calculate the discriminant (D = b² - 4ac)
- Take the square root and solve for x
Remember "FACT" to never forget the steps! ✔️
Mission: Master This Topic!
Reinforce what you learned with fun activities
Ready to Battle? Test Your Knowledge!
Practice MCQs, build combos, climb the leaderboard!
Start Practice