What is the quadratic formula and how is it derived?

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What is the quadratic formula and how is it derived?

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Answer 1 · 2015-10-25T20:32:26

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

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

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

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What is the quadratic formula and how is it derived?

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