class-10-maths-quadratic-equations

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Quadratic Equations - Class 10 Notes:

A quadratic equation is a second-degree polynomial equation in a single variable, typically written in the form: ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.

General Form:

ax^2 + bx + c = 0

Solution of Quadratic Equation:

  1. Using the Quadratic Formula:

    2. The solutions of a quadratic equation ax^2 + bx + c = 0 are given by the quadratic formula:

    x = (-b ± √(b^2 - 4ac)) / 2a

    If the discriminant (b^2 - 4ac) is:

    • Positive: The equation has two distinct real roots.
    • Zero: The equation has one real root (a repeated root).
    • Negative: The equation has two complex roots (conjugate pairs).
  2. Factoring Method:

    3. Some quadratic equations can be factored to find the roots. For example, if the equation is of the form: (x - p)(x - q) = 0, then the roots are x = p and x = q.

Examples:

1. Solve the quadratic equation: 2x^2 - 5x + 2 = 0

Solution: Using the quadratic formula,

a = 2, b = -5, c = 2

x = (-(-5) ± √((-5)^2 - 4 * 2 * 2)) / (2 * 2)

x = (5 ± √(25 - 16)) / 4

x = (5 ± √9) / 4

x = (5 ± 3) / 4

Roots: x = (5 + 3) / 4 = 2

x = (5 - 3) / 4 = 1/2

2. Solve the quadratic equation: x^2 + 6x + 9 = 0

Solution: The given equation is already factored:

(x + 3)(x + 3) = 0

Roots: x = -3 (repeated root)

Shortcut Methods to Find Roots of Quadratic Equations - Class 10 Notes:

In some cases, quadratic equations can be easily solved using shortcut methods without directly applying the quadratic formula. Here are two common shortcut methods to find the roots of quadratic equations:

1. Factorization Method:

If the quadratic equation can be factored into two binomial expressions, setting each factor to zero will give the roots.

Example:

Find the roots of the quadratic equation: x^2 - 5x + 6 = 0

Solution: The given equation can be factored as: (x - 2)(x - 3) = 0

Setting each factor to zero and solving for x:

x - 2 = 0 => x = 2

x - 3 = 0 => x = 3

So, the roots of the quadratic equation are x = 2 and x = 3.

2. Completing the Square Method:

By transforming the quadratic equation into a perfect square trinomial, the roots can be found.

Example:

Find the roots of the quadratic equation: x^2 + 6x + 9 = 0

Solution:

Step 1: Rewrite the equation as (x^2 + 6x) + 9 = 0

Step 2: Add and subtract the square of half of the coefficient of x (i.e., (6/2)^2 = 9) inside the parentheses:

(x^2 + 6x + 9 - 9) = 0

Step 3: Factor the perfect square trinomial:

(x + 3)^2 = 0

Step 4: Take the square root of both sides and solve for x:

x + 3 = 0 => x = -3

So, the root of the quadratic equation is x = -3.