How to Solve Quadratic Equation?

Let’s assume a quadratic equation P(x) = 0. The points which satisfy this equation are called solutions or roots of this quadratic equation.

These are the four common methods to find the solutions of a quadratic equation: 

1. Factorization Method
2. Completing the Square Method
3. Using the Quadratic Formula
4. Graphical Method

Let’s look at all these methods one by one through examples. 

Factorization Method

Below is steps of solving Quadratic equations Using Factorization method:

Step 1: Find two numbers such that the product of the numbers is ‘ac’ and the sum is ‘b’.

Step 2: Then write x coefficient as the sum of these two numbers and split them such that you get two terms for x.

Step 3: Factor the first two as a group and the last two terms as another group.

Step 4: Take common factors from these and on equating the two expressions with zero after taking common factors and rearranging the equation we get the roots.

Let’s consider an example of this for better understanding.

Example: Find out the solutions of the given quadratic equation using the factorization method. 

2x² – 3x + 1 = 0

Solution:

Given, 2x² – 3x + 1 = 0

⇒ 2x² – 2x – x + 1 = 0

⇒ 2x(x – 1) – 1(x -1) = 0

⇒ (2x – 1)(x-1) = 0

Now this equation is zero when either of these two terms or both of these terms are zero

So, 

Putting 2x – 1 = 0, we get x = 1/2

Similarly, x – 1 = 0, we get x = 1 

Thus, we get two roots x = 1 and 1/2

Completing Square Method

Any equation ax² + bx + c = 0 can be converted in the form (x + m)² – n² = 0. After this, we take the square roots and get the roots of the equation.

Completing the square is just a way to readjust the given quadratic equation in such a way that they come in the form of squares. Let’s see this through an example.

Example: Find the root of the given equation through complete the square method. 

x² + 4x – 5 = 0

Solution:

Given, x² + 4x – 5 = 0 

Solving by Completing Square Method

x² + 4x – 5 = 0

⇒ x² + 4x + 4 – 9 = 0

⇒ (x + 2)² – 3² = 0

⇒ (x + 2)² = 3²

Taking square root both sides, 

x + 2 = 3 and x + 2 = -3 

⇒ x = 3 -2 and x = -3 -2

⇒ x = 1 and x = -5

We have already discussed how to solve quadratic equations using the Quadratic Formula.

Graph Method

Let us suppose the general form of the quadratic equation is ax² + bx + c = 0, where a ≠ 0. The quadratic equation is a polynomial equation of degree 2, so it comes under the conic section.

Further simplifying the standard form of quadratic equation,

 y = ax² + bx + c 

⇒ y = a[(x + b/2a)² – (D/4a²)]

⇒ y – D/4a = a[(x + b/2a)²]

This resembles a parabola and we can easily draw its curve. The points where this curve cut the x-axis are the roots of the quadratic equation (or zeroes of the quadratic polynomial).

A Quadratic equation is a second-degree equation that can be represented as ax² + bx + c = 0. In this equation, x is an unknown variable, a, b, and c are constants, and a is not equal to 0. To solve it, you can use methods such as factoring, completing the square, or the quadratic formula. Each method helps find the values of x that satisfy the equation.

Let’s learn how to solve Quadratic Equations using different methods in detail.