🎯 What Exactly Is a Quadratic Equation?

A quadratic equation in one variable is an equation of the form:

• $a$, $b$, and $c$ are real numbers (called coefficients).
• $x$ is the variable.
• The highest power of $x$ is 2, so it’s called “quadratic” (from “quadratus” meaning “square”).

In CBSE Class 10 Maths, you must be able to:

• Recognise quadratic equations.
• Convert word problems into quadratic equations.
• Solve them by:
  • Factorisation
  • Completing the square (helpful for JEE later)
  • Using the quadratic formula

📋 Quick Concept Snapshot

🔍 Recognising Quadratic Equations in Different Forms

Not every quadratic equation will already look like $ax^2 + bx + c = 0$. You often need to rearrange.

Example forms:

• $3x^2 - 5x + 2 = 0$ → already in standard form
• $2x(x - 3) = 5$ → must expand and bring all terms to one side
• $(x - 2)(x + 4) = 0$ → factor form, can be expanded or solved directly

First step in MOST questions: bring everything to one side so that the right-hand side is 0.

🧠 Method 1: Solving by Factorisation (Class 10 Favourite)

You’ve probably seen this the most in your NCERT.

Example 1: Basic factorisation

Solve:

Step 1: Identify $a$, $b$, $c$.

Here, $a = 1$, $b = -5$, $c = 6$.

Step 2: Find two numbers whose:

• Product = $a \times c = 1 \times 6 = 6$
• Sum = $b = -5$

Numbers: -2 and -3 (because -2 × -3 = 6 and -2 + -3 = -5)

Step 3: Split the middle term using -2 and -3.

Step 4: Group and factor.

Step 5: Apply zero product rule.

If $(x - 3)(x - 2) = 0$, then either

• $x - 3 = 0$ so $x = 3$, or
• $x - 2 = 0$ so $x = 2$

Roots: 2 and 3.

Example 2: Factorisation when $a \ne 1$

Solve:

Step 1: Identify $a$, $b$, $c$.

$a = 6$, $b = 11$, $c = -10$

Step 2: Find two numbers whose:

• Product = $a \times c = 6 \times (-10) = -60$
• Sum = $b = 11$

Numbers: 15 and -4 (15 × -4 = -60 and 15 + (-4) = 11)

Step 3: Split the middle term.

Step 4: Group and factor.

Step 5: Apply zero product rule.

• $3x - 2 = 0$ → $x = \dfrac{2}{3}$
• $2x + 5 = 0$ → $x = -\dfrac{5}{2}$

So, the roots are $2/3$ and $-5/2$.

🧮 Method 2: Quadratic Formula (Your All-Purpose Weapon)

Sometimes factorisation is difficult or impossible (with irrational roots). Then the quadratic formula is the fastest and most reliable method.

For $ax^2 + bx + c = 0$, the roots are given by:

The expression under the square root,

is called the discriminant.

Example 3: Using the quadratic formula

Solve:

Step 1: Write $a$, $b$, $c$.

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

Step 2: Calculate the discriminant.

Step 3: Apply the quadratic formula.

So we get two roots:

• $x = \dfrac{7 + 5}{4} = \dfrac{12}{4} = 3$
• $x = \dfrac{7 - 5}{4} = \dfrac{2}{4} = \dfrac{1}{2}$

Roots are 3 and 1/2.

🔍 Discriminant and Nature of Roots (Very Exam-Friendly)

The discriminant $\Delta = b^2 - 4ac$ tells you how many and what kind of solutions the quadratic equation has.

• If $\Delta > 0$ → Two distinct real roots
• If $\Delta = 0$ → Real and equal roots (repeated root)
• If $\Delta < 0$ → No real roots, only complex roots (beyond Class 10, but useful for JEE later)

Example 4: Just checking nature of roots

For $x^2 - 4x + 4 = 0$:

Here, $a = 1$, $b = -4$, $c = 4$

So, the equation has real and equal roots.

(If you solve, you’ll find the root is 2 and 2.)

🎯 Concept Focus Box – “Roots” vs “Zeros” vs “Solutions”

• Roots of equation: Values of $x$ that satisfy $ax^2 + bx + c = 0$
• Zeros of quadratic polynomial: Values of $x$ for which $ax^2 + bx + c = 0$ (same idea from polynomial chapter)
• Solutions: Another word for roots

In Class 10 CBSE Maths, these words are often used interchangeably. In JEE-type questions, they may say “zeros” more often.

💼 Real-Life Connection: Quadratics in Motion and Money

1. Projectile motion (Physics / NEET relevance):
   Height of a ball thrown upwards is often given by an equation like
   $h(t) = -5t^2 + 20t + 1$.
   Setting $h(t) = 0$ and solving gives the times when the ball hits the ground.

2. Profit and area problems (Commerce & JEE relevance):

   • Maximum area of a rectangle with fixed perimeter leads to quadratic expressions.
   • Profit functions in business can be quadratic, where roots show break-even points.

These real-life links make quadratic equations powerful beyond just exams.

🧩 Word Problem Example (Highly Scoring in Boards)

Example 5: Number problem

“The product of two consecutive positive integers is 132. Find the integers.”

Step 1: Assume the numbers.

Let the first integer be $n$.
Then the next consecutive integer is $n + 1$.

Step 2: Write equation using the given condition.

Product = 132

Step 3: Solve the quadratic.

Find two numbers with:

• Product = $-132$
• Sum = 1

Numbers: 12 and -11 (12 × -11 = -132, 12 + (-11) = 1)

Split the middle term:

Group:

So,

• $n - 11 = 0$ → $n = 11$
• $n + 12 = 0$ → $n = -12$ (reject as numbers are positive)

So, the integers are 11 and 12.

⚠️ Common Mistakes That Cost Marks

1. Forgetting to rearrange to 0

Students often start applying formula/factorisation directly to equations like:

• $x^2 + 5x = 3$

You must first write:

2. Sign mistakes in discriminant

For $b^2 - 4ac$, carefully include the sign of $b$:

• If $b = -7$, then $b^2 = (-7)^2 = 49$, not -49.

3. Dividing by 2a incorrectly

In the quadratic formula,

The entire numerator must be divided by $2a$. Do not do:

• $-b/2a \pm \sqrt{\Delta}$ (this is wrong).

4. Skipping final statement

In CBSE board answers, always end with a clear statement like:

• “Therefore, the roots of the equation are 2 and 3.”
• “Hence, the required numbers are 11 and 12.”

This makes your solution neat and exam-ready.

📝 Exam Strategy Corner: How to Score 5/5 on Quadratic Equation Questions

1. For 1-mark questions (MCQs / very short):

   • Quickly use discriminant to check nature of roots.
   • Memorise special forms like $(x - a)^2 = 0$ gives equal roots.

2. For 2-mark questions:

   • Use factorisation method where possible (faster and cleaner).
   • Avoid long calculations; choose smart method.

3. For 3–4 mark questions:

   • Show all steps clearly.
   • Mention $a$, $b$, $c$ before using the quadratic formula.
   • Box or underline the final roots.

4. For word problems:

   • Define variables clearly: “Let the required number be x”.
   • Write equation step with a short reason: “According to the condition…”
   • Don’t forget to reject meaningless roots (like negative length, negative time).

🧾 Mini Revision Sheet – Quadratic Equations (Class 10) ✅

• Standard form: $ax^2 + bx + c = 0$, $a \ne 0$
• Roots: Values of $x$ that satisfy the equation
• Factorisation method:
  • Find numbers with product $ac$ and sum $b$
  • Split middle term, group, factor
• Quadratic formula:

• Discriminant:

• Nature of roots:

  • $\Delta > 0$ → two distinct real roots
  • $\Delta = 0$ → real and equal roots
  • $\Delta < 0$ → no real roots (complex roots)

• Always:

  • Rearrange to get 0 on one side
  • Check calculations carefully
  • Present final answer clearly

🧠 Did You Know? (For Curious Minds and Future Aspirants)

• In coordinate geometry, every quadratic equation corresponds to a parabola.
• The roots of the quadratic are the x-intercepts of this parabola.
• In JEE-level maths, you’ll study how the sign of the discriminant tells you whether the parabola cuts the x-axis at two points, touches it, or does not touch it at all.

Understanding this in Class 10 makes advanced math much easier later!

🎓 Ready to Practise “Quadratic Equations”?

Consistent practice is the secret to mastering this chapter for CBSE boards and building a strong base for JEE and NEET maths.