The reading material provided on this page for Quadratic Equations is specifically designed for students in grades 9 to 12. So, let's begin!

What is a Quadratic Equation?

A quadratic equation is a type of polynomial equation that has the form of ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. In other words, A quadratic equation is a second-degree polynomial equation in a single variable. Quadratic equations are commonly used in physics, engineering and other fields to model real-world phenomena.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is:

where a, b and c are constants and a ≠ 0.

Some examples of the quadratic equation:

x^{2} + 5x + 6 = 0

2x^{2} - 3x - 2 = 0

4x^{2} + 7x - 3 = 0

x^{2} - 9 = 0

3x^{2} + 2x + 1 = 0

In each of these examples, the highest power of the variable (x) is 2 and the equation is in the form ax^{2} + bx + c = 0, where a, b and c are constants.

Quadratic Formula

The quadratic formula offers a convenient and efficient approach to determining the roots of a quadratic equation. It is particularly useful when factorizing the equation is challenging. This formula is also known as the Sridharacharya formula.

The roots of a quadratic equation ax^{2} + bx + c = 0 are given by

How to Solve Quadratic Equation?

To find the roots of a quadratic equation, there are four distinct methods available. These methods provide various approaches to solve quadratic equations and obtain the values of "x" or the roots.

The three methods used to solve quadratic equations include:

Factorization

Completing the square

Quadratic formula

Factorization of Quadratic Equation:

Step 1: Identify two numbers that multiply to give the product of the coefficients of the x^{2} and constant terms and add up to the coefficient of the x term.

Step 2: Group the terms in pairs and factor them separately.

Step 3: Look for a common factor in each pair and factor it out.

Step 4: Write the factored form of the quadratic expression.

Which can be understood by

x^{2} + (a + b) x + ab = 0

x^{2} + ax + bx + ab = 0 x (x + a) + b (x + a) (x + a) (x + b) = 0

Here is an example to understand the factorization process.

Rewrite the left side as a perfect square trinomial:

(x - 3)^{2} = 2

Take the square root of both sides:

x - 3 = ± √2

Solve for x:

For the positive root: x - 3 = √2

x = 3 + √2

For the negative root: x - 3 = - √2

x = 3 - √2

Therefore, the quadratic equation x^{2} - 6x + 9 = 0 has two distinct real roots: x = (3 + √2) and x = (3 - √2)

Roots of Quadratic Equation

A quadratic equation is typically written in the form: ax^{2} + bx + c = 0, where "a", "b" and "c" are constants and "a" is not equal to zero. The roots of a quadratic equation are the values of "x" that satisfy the equation and make it balanced, with zero as the result when the equation is evaluated. A quadratic equation can have a maximum of two distinct roots.

For example, if we have the quadratic equation

x^{2} - 5x + 6 = 0,

its roots are the values of "x" that satisfy the equation. By factoring or using other methods, we can determine that the roots of this equation are x = 2 and x = 3. When we substitute these values back into the equation, we get:

Thus, both x = 2 and x = 3 are the roots of the quadratic equation x^{2} - 5x + 6 = 0 because they satisfy the equation and make it equal to zero.

Nature of Roots

A quadratic equation can have a maximum of two roots. These roots can be real and distinct (two different real numbers), real and equal (one repeated real number), or complex (involving imaginary numbers).

Discriminant

The discriminant is used in quadratic equations to determine the nature and number of its roots.

D = (b^{2} - 4ac)

where "a", "b" and "c" are the coefficients of the quadratic equation (ax^{2} + bx + c = 0).

D > 0, the roots are real and distinct

D = 0, the roots are real and equal.

D < 0, the roots do not exist or the roots are imaginary.

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