How do you derive the quadratic formula? Thanks

1 Answer
Jan 30, 2018

See below.

Explanation:

The #color(blue)"Quadratic Formula"# can derived by completing the square algebraically.

From the form:

#ax^2+bx+c#

Start with:

#ax^2+bx+c=0#

Move the constant #c# to the left-hand side:

#ax^2+bx=-c#

Divide by the coefficient of #x^2#:

#a/ax^2+b/ax=-c/a#

#x^2+b/ax=-c/a#

Add to both sides the square of half the coefficient of #x#:

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

Simplify left hand side:

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

#x^2+b/ax+b/(2a)=(b^2-4ac)/(4a^2)#

Arrange right hand side into the square of a binomial:

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

Take square roots of both sides:

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

Subtract #b/(2a)# from both sides:

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

Add fractions:

#color(blue)(x=(-b+-sqrt((b^2-4ac)))/(2a))#