How to Solve Quadratic Equations?

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:

1. x² + 5x + 6 = 0
2. 2x² - 3x - 2 = 0
3. 4x² + 7x - 3 = 0
4. x² - 9 = 0
5. 3x² + 2x + 1 = 0

In each of these examples, the highest power of the variable (x) is 2 and the equation is in the form ax² + 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 ax² + 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:

1. Factorization
2. Completing the square
3. Quadratic formula

1. Factorization of Quadratic Equation:

Step 1: Identify two numbers that multiply to give the product of the coefficients of the x² 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

x² + (a + b) x + ab = 0

x² + ax + bx + ab = 0
x (x + a) + b (x + a)
(x + a) (x + b) = 0

Here is an example to understand the factorization process.

x² + 11x + 30 = 0

x² + 5x + 6x + 30 = 0
x (x + 5) + 6 (x + 5) = 0
(x + 5) (x + 6) = 0

2. Quadratic Formula: Steps to solve a quadratic equation using the quadratic formula:

Step 1: Start with a quadratic equation in the form ax^2 + bx + c = 0.

Step 2: Plug the values of “a,” “b,” and “c” into the quadratic formula:

Step 3: Simplify the formula by calculating the discriminant: (b² - 4ac).

Step 4: Determine the nature of the roots based on the value of the discriminant.

Here is an example to understand the factorization process.

2x² + 5x - 3 = 0

Where a = 2, b = 5 and c = -3, on substituting in formula

 x = [ (-b) ± √ (b² - 4ac)] / (2a).

For the positive root: x = (-5 + 7) / 4 = 2 / 4 = 1/2.

For the negative root: x = (-5 - 7) / 4 = -12 / 4 = -3.

Therefore, the quadratic equation 2x² + 5x - 3 = 0 has two distinct real roots: x = 1/2 and x = -3.

3. Method of Completing the Square: By completing the square, we can find the roots of the equation.

Start with the quadratic equation: x² - 6x + 7 = 0.

Move the constant term (7) to the other side of the equation:

x² - 6x = -7.

To complete the square, take half of the coefficient of x (-5/2), square it [(6/2)² = 9] and add it to both sides of the equation:

x² - 6x + 9 = -7 + 9
x² - 6x + 9 = 2
x² - 6x + 9 = 2

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 x² - 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: ax² + 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

x² - 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:

(2)² - 5(2) + 6 = 0 => 4 - 10 + 6 = 0 => 0 = 0 (True)

(3)² - 5(3) + 6 = 0 => 9 - 15 + 6 = 0 => 0 = 0 (True)

Thus, both x = 2 and x = 3 are the roots of the quadratic equation x² - 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 = (b² - 4ac)

where "a", "b" and "c" are the coefficients of the quadratic equation (ax² + 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.