Question

What is the quadratic formula and how is it derived?

Answer 1

For any general quadratic equation of the form `ax^2+bx+c=0`, we have the quadratic formula to find the values of x satisfying the equation and is given by
`x=(-b+-sqrt(b^2-4ac))/(2a)`

To derive this formula, we use completing the square in the general equation `ax^2+bx+c=0`

Dividing throughout by a we get : `x^2+b/ax+c/a=0`

Now take the coefficient of x, half it, square it, and add it to both sides and rearrange to get

`x^2+b/ax+(b/(2a))^2=b^2/(4a)^2-c/a`

Now right the left hand side as a perfect square and simplify the right hand side.

`therefore(x+b/(2a))^2=(b^2-4ac)/(4a^2)`

Now taking the square root on both sides yields :

`x+b/(2a)=+-sqrt((b^2-4ac))/(2a)`

Finally solving for x gives

`x=-b/(2a)+-sqrt(b^2-4ac)/(2a)`

`x=(-b+-sqrt(b^2-4ac))/(2a)`