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Home >> Polynomials >> Quadratic Equation >> Finding roots of Quadratic Equation >>

The formula to find the roots of a quadratic equation is known as the Quadratic Formula:

Note: if b2 - 4ac > 0 then only we can find the roots of quadratic equation with this formula.

Let's understand how we get this formula

There are two methods in this:

Method 1 : Completing the Square

ax2 + bx + c = 0

Note : a ≠ 0

Divide both sides by "a" and we get:

 x2 + b a x + c a = 0

or we can also write it as

 x2 + b a x + ⎧ ⎩ b 2a ⎫ 2 ⎭ x - ⎧ ⎩ b 2a ⎫ 2 ⎭ x + c a = 0

Now, here see the first three variable i.e

 x2 + b a x + ⎧ ⎩ b 2a ⎫ 2 ⎭

You will notice that here (a + b)2 formula applies and we get:

 ⎧ ⎩ x + b 2a ⎫ 2 ⎭ - ⎧ ⎩ b 2a ⎫ 2 ⎭ + ⎧ ⎩ c a ⎫ ⎭ = 0

 solving ⎧ ⎩ b 2a ⎫2 ⎭ we get:

 ⎧ ⎩ x + b 2a ⎫ 2 ⎭ - ⎧ ⎩ b 2 4a ⎫ ⎭ + ⎧ ⎩ c a ⎫ ⎭ = 0

 ⎧ ⎩ x + b 2a ⎫ 2 ⎭ - ⎧ ⎩ b 2 - 4ac 4a2 ⎫ ⎭ = 0

 Move - ⎧ ⎩ b 2 - 4ac 4a2 ⎫ ⎭ to RHS and we get

 ⎧ ⎩ x + b 2a ⎫ 2 ⎭ = ⎧ ⎩ b 2 - 4ac 4a2 ⎫ ⎭

Take square root of both sides and we get:

 x + b 2a = ± √ ( b 2 - 4ac ) 2a2

 Move ⎧ ⎩ b 2a ⎫ ⎭ to RHS we get:

 x = - ⎧ ⎩ b 2a ⎫ ⎭ ± √ ( b 2 - 4ac ) 2a2

On joining like terms we get:

 x = -b ± √ ( b 2 - 4ac ) 2a2

So, the roots of given equation is :

 x = -b + √ ( b 2 - 4ac ) 2a2 or x = -b - √ ( b 2 - 4ac ) 2a2

Method 2: It is a quite old and shorter method being used by Indian Mathematicians:

ax2 + bx + c = 0

multiply both sides with 4a and we get:

4a2 x2 + 4abx + 4ac = 0

move 4ac on the RHS and we get:

4a2 x2 + 4abx = - 4ac

add b2 on both sides and we get:

4a2 x2 + 4abx + b2 = b2 - 4ac

You will notice that on LHS (a + b)2 formula applies and we get:

(2ax + b)2 = b2 - 4ac

Taking square root on both sides and we get:

 2ax + b = ± √ ( b 2 - 4ac )

Subtract b from side and we get:

 2ax = -b ± √ ( b 2 - 4ac )

Divide both sides by 2a and we get:

 x = -b ± √ ( b 2 - 4ac ) 2a

Now let's apply the above explained formula and find solution for the following quadratic equation:

Example : 2x2 -5x + 3 = 0

Solution: Find the values of a, b and c in the given equation and we get:
a = 2
b = -5
c = 3

Put the above values in above calculated quadratic formula and we get

 x = -(-5) ± √ ( -5 2 - 4 x 2 x 3 ) 2 x 2

 x = 5 ± √ ( 25 - 24 ) 4

 x = 5 ± √ 1 4

Calculating square root and we get:

 x = 5 ± 1 4

 x = 5 + 1 4 = 6 4 = 3 2

 or x = 5 - 1 4 = 4 4 = 1

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