Finding roots of Quadratic Equation or solution of quadratic equation by quadratic formula

Standard Form of Quadratic Equation

Finding roots of Quadratic Equation

Discriminant 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 Consider the following quadratic formula: 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: Consider the following quadratic formula: 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