How to find the roots of the quadratic equation by 3 methods? | TIRLA ACADEMY

There are various equations in mathematics such as linear, Quadratic, Cubic, etc.

The degree of the polynomial is one so it is a linear equation, if the degree of the polynomial is two so it is quadratic, and if the degree of the polynomial is three so it is a cubic equation.

In this article, we are going to talk about Quadratic equations and how to find the roots of quadratic equations by 3 methods?

The topics related to Quadratic Equation that we are covering-

1. What is the Quadratic equation definition in math?

2. What is the general form of a Quadratic Equation?

3. What are the examples of Quadratic Equations?

4. What are the methods to find the Roots of Quadratic Equations?

5. How to solve a quadratic equation by factoring?

6. How to solve a quadratic equation by completing the square method?

7. How to find quadratic equation solutions by Quadratic Formula?

8. FAQs on Quadratic Equation

So let's discuss what is a quadratic equation?

1. What is the Quadratic equation definition in math?

An equation whose higher degree of a polynomial is 2 is known as a Quadratic equation. 

Here is a chart showing various types of equations are-

There are various terms in the quadratic equations but it has also a standard form. Let's understand-

2. What is the general form of a Quadratic Equation?

The general form of a quadratic equation is ax²+bx+c=0.

Where,
x = variable
a = coefficient of x²
b = coefficient of x
c = constant term

But remember, all quadratic equation does not look like a general form. Here are some quadratic equation examples.

3. What are the examples of Quadratic Equations?

1.) 2x²+5x+3
2.) 5y²-5
3.) 7 + x² + 6x
4.) -3t + 2t² + 1

5.) p²

Now, discuss the various methods to find the roots or solutions of quadratic equations.

4. What are the methods to find the Roots of Quadratic Equations?

We use three methods to find the roots of the Quadratic Equations. They are-

1.) Factorisation method or Splitting method
2.) Completing the Square method
3.) Quadratic Formula method

Now let us discuss how to find the roots of a Quadratic equation by various methods?

5. How to solve a quadratic equation by factoring?

In the factorisation method, we do factors of Quadratic Equation or split the middle term and find the roots.

But how it works let's take an example-

⟹2x² + 5x + 3 = 0

⟹2x² + 2x + 3x + 3 = 0

⟹2x(x + 1) + 3(x + 1) = 0

⟹(x + 1) (2x + 3) = 0

⟹(x + 1) = 0, (2x + 3) = 0

⟹x = -1 , x = -3/2

But how does completing the square method work to find the roots?

6. How to solve a quadratic equation by completing the square method?

In Completing the Square method we use Algebraic Identities of Squares to find the roots or solutions of Quadratic Equations.

Now take an example to understand it-

⟹2x² + 5x + 3 = 0

⟹2(2x² + 5x + 3) = 0

⟹4x² + 10x + 6 = 0

⟹(2x)² + 2×2x×(5/2)+(5/2)²-(5/2)²+6=0

⟹(2x + 5/2)² - (5/2)² + 6 = 0

⟹(2x + 5/2)² - 25/4 + 6 = 0

⟹(2x + 5/2)² - (25 + 24)/4 = 0

⟹(2x + 5/2)² - 1/4 =0

⟹(2x + 5/2)² = 1/4

⟹(2x + 5/2) = √(1/4)

⟹2x + 5/2 = ±1/2

⟹2x = ±1/2 - 5/2

Taking positive sign

⟹2x = +1/2 - 5/2

⟹2x = (1-5)/2

⟹2x = -4/2

⟹x = -2/2

⟹x = -1

Taking negative sign

⟹2x = -1/2 - 5/2

⟹2x = (-1-5)/2

⟹2x = -6/2

⟹x = -3/2

Now, it turns to understanding the quadratic equation formula method.

7. How to find quadratic equation solutions by Quadratic Formula?

As per its name, we use the quadratic equation formula to find the solution or roots of a quadratic equation.

Take an example to understand it-

2x² + 5x + 3 = 0

where, a = 2, b = 5, c = 3

Put all the values in the Quadratic Formula-

x = {-b ± √(b² - 4ac)}÷(2a)

x = {-5 ± √(5² - 4×2×3)}÷(2×2)

x = {-5 ± √(25 - 24)} ÷ 4

x = {-5 ± √1} ÷ 4

x = (-5 ± 1)/4

Taking positive sign

x = (-5 + 1)/4

x = -4/4

x = -1

Taking negative sign

x = (-5 - 1)/4

x = -6/4

x = -3/2

FAQs on Quadratic Equation

When quadratic equation has equal roots?

A quadratic equation has equal roots if b²-4ac=0.

When a quadratic equation has no solution?

A quadratic equation has no solution if b²-4ac<0.

Why quadratic equation is called quadratic?

Because the degree of a polynomial is 2 so the equation is known as a Quadratic equation.

What is a quadratic equation as the sum and product of roots?

The sum of roots is α+β = (-b/a) and the product of roots is αβ = (c/a).

Quadratic equation vs linear equation

If the degree of a polynomial is 1 so the equation is linear and if the degree of a polynomial is 2 so the equation is quadratic.

What is a Quadratic equation in standard form?

The standard form of a quadratic equation is ax²+bx+c=0.