Quadratic Equations Class 10: Score Full 6 Marks

The fastest method when the equation factorises cleanly. We'll show how to spot factorable equations in 5 seconds.

Factorisation works when you can break down the quadratic into two linear factors. For example, x² + 5x + 6 = 0 factors as (x + 2)(x + 3) = 0, giving roots x = -2 and x = -3.

The 5-second spotting trick: Check if the coefficient of x² is 1. Then look for two numbers that multiply to give the constant term and add to give the coefficient of x. If you find them instantly, factorise. If not, skip to Method 2.

Example from CBSE papers: x² - 7x + 12 = 0. The two numbers are –3 and –4 (they multiply to 12 and add to –7). So it factors as (x - 3)(x - 4) = 0.

The safety net. Works on every quadratic. But there's one substitution mistake that 60% of students make under exam pressure.

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

For ax² + bx + c = 0, you must identify a, b, and c correctly—including their signs. This is where most students slip.

The common mistake: Writing b = 5 when the equation is 2x² - 5x + 3 = 0. The correct values are a = 2, b = -5, and c = 3. Notice b is negative.

Working it through: x = [5 ± √(25 - 24)] / 4 = [5 ± 1] / 4, so x = 1.5 or x = 1.

Always rewrite the equation in standard form ax² + bx + c = 0 before you identify the coefficients.

CBSE asks this directly in 2-mark questions. Most students panic. It's actually the simplest method when you know the pattern.

The idea: rewrite x² + bx + c = 0 as a perfect square.

The pattern: Divide the coefficient of x by 2, square it, then add and subtract.

For x² + 6x - 7 = 0:

• Coefficient of x is 6. Half of it is 3. Square it: 9.
• Rewrite: x² + 6x + 9 - 9 - 7 = 0
• Simplify: (x + 3)² - 16 = 0
• Solve: (x + 3)² = 16, so x + 3 = ±4, giving x = 1 or x = -7.

This method builds intuition for why roots exist and is often the fastest for mental calculation.

Every quadratic word problem reduces to the same 3 steps. Once you see them, train problems and area problems feel identical.

Step 1: Identify the unknown. Let it be x.

Step 2: Translate the English into an equation using the given information.

Step 3: Solve using Method 1, 2, or 3. Discard any root that doesn't make sense (e.g., negative length).

Example (CBSE typical): "A rectangular garden is 4 m longer than it is wide. Its area is 96 m². Find its dimensions."

• Let width = x m. Then length = x + 4 m.
• Area: x(x + 4) = 96 → x² + 4x - 96 = 0
• Factorise: (x + 12)(x - 8) = 0 → x = -12 or x = 8
• Since width cannot be negative, x = 8 m and length = 12 m.

When the question says "find the nature of roots", they're testing one thing: do you know what b² - 4ac tells you?

The discriminant Δ = b² - 4ac determines:

• If Δ > 0: Two distinct real roots
• If Δ = 0: One repeated real root (or two equal roots)
• If Δ < 0: No real roots (imaginary roots exist)

Practical takeaway for exams: Before solving any quadratic, compute the discriminant first. It tells you how many roots to expect and what type. This saves you from careless errors and builds confidence. Memorise these three conditions tonight.

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