Question

Complete the steps of derivation of the quadratic formula

Answer 1

Step 1. We are going to begin with standard form a quadratic equation: $ax^2+bx+c=0$

Step 2.  Divide both sides of the equation by $a$:
$\frac{ax^2+bx+c}{a} = \frac{0}{a}$
$\frac{ax^2}{a} + \frac{bx}{c} + \frac{c}{a} =0$
$x^{2} + \frac{b}{a} x+ \frac{c}{a} =0$

Step 3. Subtract $\frac{c}{a}$ from both sides of the equation:
$x^{2} + \frac{b}{a} x+ \frac{c}{a}- \frac{c}{a} =0 - \frac{c}{a}$
$x^{2} + \frac{b}{a} x=- \frac{c}{a}$

Step 4. Complete the square of the left hand side by adding $( \frac{b}{2a} )^2$ to both sides:
$x^{2} + \frac{b}{a} x+( \frac{b}{2a} )^2=- \frac{c}{a} +( \frac{b}{2a} )^2$
$(x+ \frac{b}{2a} )^2=- \frac{c}{a} +( \frac{b}{2a} )^2$

Step 5. Take square root to both sides of the equation:
$\sqrt{(x+ \frac{b}{2a} )^2} =+/- \sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}$
$x+ \frac{b}{2a}=+/-\sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}$

Step 6. Subtract $\frac{b}{2a}$ from both sides of the equation:
$x+ \frac{b}{2a}- \frac{b}{2a} =- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}$
$x=- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}$

Step 7. Simplify the radicand of the left hand side of the equation:
$x=- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} + \frac{b^2}{(2a)^2}}$
$x=- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} + \frac{b^2}{4a^2}}$
$x=- \frac{b}{2a} +/-\sqrt{- \frac{4ac}{4a^2} + \frac{b^2}{4a^2}}$
$x=- \frac{b}{2a} +/-\sqrt{- \frac{4ac+b^2}{4a^2} }$

Step 8. Take $4a^2$ outside the radical:
$x=- \frac{b}{2a} +/- \frac{ \sqrt{-4ac+b^2} }{ \sqrt{4a^2} }$
$x=- \frac{b}{2a} +/- \frac{ \sqrt{-4ac+b^2} }{2a} }$

Step 9. Combine the two fractions and rearrange the terms in the radical: 
$x= \frac{-b+/- \sqrt{b^2-4ac} }{2a}$