Quadratic Equations

Overview

Quadratic equations form a cornerstone of algebra and appear frequently in MP TET Varg-2 Mathematics. A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. These equations model numerous real-world situations—projectile motion, area calculations, profit-loss problems, and more.

For the MP TET exam, you must master identifying quadratic equations, finding their roots using multiple methods, understanding the nature of roots through the discriminant, and applying the relationship between roots and coefficients. Questions typically test computational accuracy and conceptual clarity, often mixing word problems with direct calculations.

This topic connects directly to algebraic expressions, factorisation, and linear equations. A solid grasp here also supports understanding of coordinate geometry and graphing parabolas at higher levels.

Key Concepts

• **Standard Form**: A quadratic equation in variable x is written as ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. The condition a ≠ 0 is crucial—without it, the equation becomes linear.

• **Roots/Solutions**: The values of x that satisfy the equation are called roots or zeros. A quadratic equation has exactly two roots (which may be equal, real, or complex).

• **Discriminant (D)**: The expression D = b² − 4ac determines the nature of roots. It tells us whether roots are real and distinct, real and equal, or imaginary—without actually solving the equation.

• **Sum and Product of Roots**: If α and β are roots of ax² + bx + c = 0, then Sum (α + β) = −b/a and Product (αβ) = c/a. These relationships help form equations when roots are known.

• **Methods of Solution**: Four primary methods exist—factorisation, completing the square, quadratic formula, and graphical method. Factorisation is quickest when applicable; the quadratic formula works universally.

• **Parabola Connection**: The graph of y = ax² + bx + c is a parabola. The roots of the equation are the x-coordinates where the parabola crosses the x-axis.

Formulas / Key Facts

| Formula/Fact | Context | |--------------|---------| | ax² + bx + c = 0 (a ≠ 0) | Standard form of quadratic equation | | D = b² − 4ac | Discriminant formula | | D > 0 → Two distinct real roots | Parabola cuts x-axis at two points | | D = 0 → Two equal real roots | Parabola touches x-axis at one point | | D < 0 → No real roots (imaginary) | Parabola doesn't touch x-axis | | x = (−b ± √D) / 2a | Quadratic formula (Shreedharacharya's rule) | | Sum of roots = −b/a | α + β = −b/a | | Product of roots = c/a | αβ = c/a | | If roots are α, β, equation is x² − (α+β)x + αβ = 0 | Forming equation from roots |

Worked Examples

**Example 1: Solving by Factorisation**

Solve: x² − 5x + 6 = 0

Step 1: Find two numbers whose product = 6 and sum = −5 Numbers are −2 and −3 (since −2 × −3 = 6 and −2 + −3 = −5)

Step 2: Factorise x² − 2x − 3x + 6 = 0 x(x − 2) − 3(x − 2) = 0 (x − 2)(x − 3) = 0

Step 3: Solve each factor x − 2 = 0 → x = 2 x − 3 = 0 → x = 3

**Roots: x = 2 and x = 3**

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**Example 2: Using Quadratic Formula**

Solve: 2x² + 7x + 3 = 0

Here a = 2, b = 7, c = 3

Step 1: Calculate discriminant D = b² − 4ac = 49 − 24 = 25

Step 2: Apply formula x = (−7 ± √25) / (2 × 2) x = (−7 ± 5) / 4

Step 3: Find both roots x = (−7 + 5)/4 = −2/4 = −1/2 x = (−7 − 5)/4 = −12/4 = −3

**Roots: x = −1/2 and x = −3**

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**Example 3: Finding Nature of Roots**

Without solving, determine the nature of roots of 3x² − 2x + 5 = 0

Here a = 3, b = −2, c = 5

D = b² − 4ac = 4 − 60 = −56

Since D < 0, the equation has **no real roots** (roots are imaginary).

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**Example 4: Forming Equation from Roots**

Form a quadratic equation whose roots are 4 and −7.

Sum of roots = 4 + (−7) = −3 Product of roots = 4 × (−7) = −28

Using x² − (sum)x + (product) = 0 **Equation: x² + 3x − 28 = 0**

Common Mistakes

• **Forgetting a ≠ 0**: Students sometimes accept a = 0 in the standard form. If a = 0, the equation bx + c = 0 is linear, not quadratic. Always verify a ≠ 0.

• **Sign errors in the quadratic formula**: The formula has −b, not b. When b is already negative, −b becomes positive. Write out each step to avoid sign confusion.

• **Incorrect discriminant calculation**: Students often compute b² − 4ac as b² − 4 × a × c incorrectly, especially with negative values. Use brackets: D = (b)² − 4(a)(c).

• **Confusing sum formula sign**: Sum of roots = −b/a (note the negative sign). Students often write b/a, forgetting the minus. Remember: "Sum has the negative."

• **Incomplete factorisation**: When factorising, students sometimes stop at x(x − 5) = 6 and solve as x = 6. This is wrong. The equation must equal zero before factorising.

• **Assuming equal roots means one root**: When D = 0, there are still two roots—they happen to be equal (repeated root). Don't say "one root."

Quick Reference

• Standard form: ax² + bx + c = 0 where a ≠ 0

• Discriminant D = b² − 4ac decides nature: positive (2 real), zero (2 equal), negative (no real)

• Quadratic formula: x = (−b ± √D) / 2a — works for every quadratic

• Sum of roots = −b/a, Product of roots = c/a

• To form equation from roots α, β: x² − (α + β)x + αβ = 0

• Factorisation is fastest but only works when roots are rational and simple