Question

derive the formula to find the ratio of quadratic equation ax²+bx+c=0​

Answer 1

$\huge\fcolorbox{black}{lime}{AnsweR:}$

The derivation of this formula can be outlined as follows:

•  Divide both sides of the equation ax2 + bx + c = 0 by a.

•  Transpose the quantity c/a to the right side of the equation.

•  Complete the square by adding b2 / 4a2 to both sides of the equation.

•  Factor the left side and combine the right side.

•  Extract the square-root of both sides of the equation.

•  Solve for x by transporting the quantity b / 2a to the right side of the equation.

•  Combine the right side of the equation to get the quadratic formula.

See the derivation below.

$\bold{ax^2+bx+c=0,}\\\\\bold{x^2+\frac{b}{a}x+\frac{c}{a}=0, }\\\\\bold{x^2+\frac{b}{a}x=-\frac{c}{a} }\\\\\bold{x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=\frac{b^2}{4a^2}-\frac{c}{a}, }\\\\\bold{(x+\frac{b}{2a} )^2=\frac{b^2-4ac}{4a^2}, }\\\\\bold{x+\frac{b}{2a}=\frac{+-\sqrt{b^2-4ac} }{2a}, }\\\\\bold{x=\frac{-b}{2a}+-\frac{\sqrt{b^2-4ac} }{2a}. }$