The quadratic formula

The quadratic formula is an alternative to solving a quadratic equation.

To solve a quadratic equation means to find roots of a quadratic equation.

Roots of a quadratic equation are values of the variable in the quadratic equation, say x, which reduce the value of the equation to zero.

Now consider a quadratic equation of the standard form:

ax^{2} + bx + c = 0

We know a quadratic equation can be solved by factoring method.

But, not all trinomials in the form of a quadratic equation can be solved by the factoring method.

Consider for example the following equation:

**x ^{2} + 4x + 2 = 0**

In this quadratic equation, the product of roots, *which have yet to be solved for,* is 1 × 2, i.e. 2.

But what two integers can be there, having a product of 2, with a sum -4 and ** ax^{2} + bx + c = 0**. None, indeed !

Therefore, the need for an alternative for finding the roots of this quadratic equation arises.

The alternative is called the “The Quadratic Formula”

When factoring a quadratic equation gets difficult, then the quadratic formula enables us to find roots of a quadratic equation of the form *ax ^{2} + bx + c = 0*

The quadratic formula is

In this formula, *x* is roots of the quadratic equation *ax ^{2} + bx + c = 0*

Derivation of the quadratic formula:

Consider the quadratic equation *ax ^{2} + bx + c = 0*

Now, ax^{2} + bx = - c

Divide both sides of the equation by a. So,

x^{2} +( b/a) x =-c/a

let us add b^{2}/4a^{2} to both sides of above, so we get

x^{2} +( b/a) x+ b^{2}/4a^{2} = -c + (b^{2}/4a^{2} )

factorize the perfect square trinomial in the left side, and apply LCM in the right

(x + (b/2a))^{2} = (b^{2} – 4ac)/4a^{2}

Take square roots on both sides,

x + (b/2a) = ± [√(b^{2} – 4ac)/2a]

Finally,

This is the solution to the general quadratic equation *ax ^{2} + bx + c*

And the solution

is called the quadratic formula.

**Example 1: **

Solve the quadratic equation x^{2} + 4x + 2 = 0

**Solution : **

As tried above, this quadratic equation cannot be solved by the factoring method.

Therefore, let us use the quadratic formula derived above.

Comparing the given quadratic equation x^{2} + 4x + 2 = 0 with the general form *ax ^{2} + bx + c*

We see, a = 1, b = 4, c = 2. Now the roots of the quadratic equation x^{2} + 4x + 2 = 0, using the quadratic formula are

Now, square root of 2 is an irrational number, therefore factoring method could not work for

finding roots of this quadratic equation.

Therefore, the two roots of the quadratic equation **x ^{2} + 4x + 2 = 0** are

**x = - 2 + √2 or x = - 2 - √2**