Derivation of Quadratic Formula

We can easily derive the quadratic formula, using two methods 

• By Completing the Square Technique
• Shortcut Method of Derivation

Now let’s learn about them in detail.

By Completing the Square Technique

Derivation of the quadratic formula is achieved by using the Completing Square method

Let us take the standard form of a quadratic equation i.e.

ax² + bx + c = 0

Dividing the equation by the coefficient of x², i.e., a.

x² + (b/a)x + (c/a) = 0

Subtracting c/a from both sides.

x² + (b/a)x = -c/a

Now, by completing the square method,

We have to add a specific constant to both sides of the equation to make the LHS a complete square.

Here, we add (b/2a)² to both sides of the equation

x² + (b/a)x + (b/2a)² = (-c/a) + (b/2a)²

Using a² + 2ab + b² = (a + b)²,

⇒ [x + (b/2a)]² = (-c/a) + (b²/4a²)

⇒ [x + (b/2a)]² = (b² – 4ac)/4a²

Taking square root on both sides,

[x + (b/2a)] = √[(b² – 4ac)] / 2a

Now, subtracting b/2a from both sides we get

x = [-b ± √(b² – 4ac)] / 2a

This is the required quadratic formula.

Shortcut Method of Derivation

The quadratic formula is derived by the shortcut method as,

The given standard form of a quadratic equation is,

ax² + bx + c = 0

Multiply both sides by 4a

4a(ax² + bx + c) = 4a × (0)

⇒ 4a²x² + 4abx + 4ac = 0

⇒ 4a²x² +4abx = -4ac

By completing the square method, add b² on both sides.

4a²x² + 4abx + b² = b² – 4ac

⇒ (2ax)² + 2(2ax)(b) + b² = b² – 4ac    {We know that, a² + 2ab + b² = (a + b)²)}

⇒ (2ax + b)² = b² – 4ac

Taking square root

2ax + b = ±√(b² – 4ac)

⇒ 2ax = -b ±√(b² – 4ac)

x = [-b ±√(b² – 4ac)]/2a

This proves the quadratic formula.

Quadratic Formula is used to find the roots (solutions) of any quadratic equation. Using the Quadratic formula real and imaginary all the types of roots of the quadratic equations are found.

The quadratic formula was formulated by a famous Indian mathematician Shreedhara Acharya, hence it is also called Shreedhara Acharya’s Formula. It is used to find the solution of the quadratic equation of the form ax² + bx + c = 0. So, let’s start learning about the concept of Quadratic Formula.