CBSE Class 10 Maths Notes Chapter 4 Quadratic Equations

CBSE Class 10 Maths Notes Chapter 4 Quadratic Equations

Quadratic Polynomial

A quadratic polynomial takes the form ax^2 + bx + c, where 'a', 'b', and 'c' represent real numbers, and 'a' is not equal to zero.

Quadratic Equation

When a quadratic polynomial is set equal to a constant, it forms a quadratic equation. Any equation expressed as p(x) = k, where p(x) represents a polynomial of degree 2 and k is a constant, falls under the category of quadratic equations.

The Standard Form of a Quadratic Equation

In the standard form of a quadratic equation, ax^2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. Here, 'a' represents the coefficient of x^2, known as the quadratic coefficient. 'b' stands for the coefficient of x, termed as the linear coefficient. Lastly, 'c' denotes the constant term.

Roots of a Quadratic Equation

The values of x that satisfy a quadratic equation are referred to as the roots of the quadratic equation. If α is a root of the quadratic equation ax^2 + bx + c = 0, then aα^2 + bα + c = 0. A quadratic equation can possess two distinct real roots, two equal roots, or real roots may not exist at all. Graphically, the roots of a quadratic equation are the points where the graph of the quadratic polynomial intersects the x-axis. Let's take the example of the graph of the quadratic equation x^2 - 4 = 0.

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Solving a Quadratic Equation by Factorization Method

Let's consider the quadratic equation 2x^2 - 5x + 3 = 0. To solve it, we split the middle term by finding two numbers (-2 and -3) such that their sum is equal to the coefficient of x and their product is equal to the product of the coefficient of x^2 and the constant. So, (-2) + (-3) = (-5) and (-2) × (-3) = 6. By splitting the middle term, we rewrite the equation as 2x^2 - 2x - 3x + 3 = 0. Then, we factorize it as 2x(x - 1) - 3(x - 1) = 0, which further simplifies to (x - 1)(2x - 3) = 0. Thus, x = 1 and x = 3/2 are the roots of the given quadratic equation. This method of solving a quadratic equation is called the factorization method.

Solving a Quadratic Equation by Completion of Squares Method

Let's apply the method of completing the square to solve the quadratic equation 2x^2 - 8x = 10: (i) Express the quadratic equation in standard form: 2x^2 - 8x - 10 = 0 (ii) Divide the equation by the coefficient of x^2 to make the coefficient of x^2 equal to 1: x^2 - 4x - 5 = 0 (iii) Add the square of half of the coefficient of x to both sides of the equation to get an expression of the form x^2 ± 2kx + k^2: (x^2 - 4x + 4) - 5 = 0 + 4 (iv) Isolate the above expression, (x ± k)^2, on the LHS to obtain an equation of the form (x ± k)^2 = p^2: (x - 2)^2 = 9 (v) Take the positive and negative square roots: x - 2 = ±3 x = -1 or x = 5

Quadratic Formula

The Quadratic Formula provides a direct method to find the roots of a quadratic equation in its standard form. For the quadratic equation ax^2 + bx + c = 0, the formula is: x = [-b ± √(b^2 - 4ac)] / (2a) By substituting the values of a, b, and c into the formula, we can determine the roots of the equation. For example, if we have the quadratic equation x^2 – 5x + 6 = 0, we can find the roots using the quadratic formula. Given: x^2 – 5x + 6 = 0 Comparing with the standard quadratic equation, we get: a = 1, b = -5, and c = 6 Since b^2 – 4ac = (-5)^2 – 4 × 1 × 6 = 25 – 24 = 1 > 0, the roots are real. Using the quadratic formula: x = [-(-5) ± √1] / (2 * 1) = [5 ± 1] / 2 = (5 + 1)/2 and (5 – 1)/2 = 6/2, 4/2 Thus, the roots of the quadratic equation are 3 and 2.