Deriving the Quadratic Formula Follow these steps to use

Chapter 0, Problem P.5.78

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Deriving the Quadratic Formula

Follow these steps to use completing the square to solve \(a x^{2}+b x+c=0\), \(a \neq 0\).

(a) Subtract c from both sides of the original equation and divide both sides of the resulting equation by a to obtain

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

(b) Add the square of one-half of the coefficient of x in (a) to both sides and simplify to obtain

\(\left(x+\frac{b}{2 a}\right)^{2}=\frac{b^{2}-4 a c}{4 a^{2}}\)

(c) Extract square roots in (b) and solve for x to obtain the quadratic formula

\(x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\)