Complete the steps of derivation of the quadratic formula
Accepted Solution
A:
Step 1. We are going to begin with standard form a quadratic equation: [tex]ax^2+bx+c=0[/tex]
Step 2. Divide both sides of the equation by [tex]a[/tex]: [tex] \frac{ax^2+bx+c}{a} = \frac{0}{a} [/tex] [tex] \frac{ax^2}{a} + \frac{bx}{c} + \frac{c}{a} =0[/tex] [tex] x^{2} + \frac{b}{a} x+ \frac{c}{a} =0[/tex]
Step 3. Subtract [tex] \frac{c}{a} [/tex] from both sides of the equation: [tex]x^{2} + \frac{b}{a} x+ \frac{c}{a}- \frac{c}{a} =0 - \frac{c}{a} [/tex] [tex]x^{2} + \frac{b}{a} x=- \frac{c}{a} [/tex]
Step 4. Complete the square of the left hand side by adding [tex]( \frac{b}{2a} )^2[/tex] to both sides: [tex]x^{2} + \frac{b}{a} x+( \frac{b}{2a} )^2=- \frac{c}{a} +( \frac{b}{2a} )^2[/tex] [tex](x+ \frac{b}{2a} )^2=- \frac{c}{a} +( \frac{b}{2a} )^2[/tex]
Step 5. Take square root to both sides of the equation: [tex] \sqrt{(x+ \frac{b}{2a} )^2} =+/- \sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2} [/tex] [tex]x+ \frac{b}{2a}=+/-\sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}[/tex]
Step 6. Subtract [tex] \frac{b}{2a} [/tex] from both sides of the equation: [tex]x+ \frac{b}{2a}- \frac{b}{2a} =- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}[/tex] [tex]x=- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} +( \frac{b}{2a} )^2}[/tex]
Step 7. Simplify the radicand of the left hand side of the equation: [tex]x=- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} + \frac{b^2}{(2a)^2}} [/tex] [tex]x=- \frac{b}{2a} +/-\sqrt{- \frac{c}{a} + \frac{b^2}{4a^2}} [/tex] [tex]x=- \frac{b}{2a} +/-\sqrt{- \frac{4ac}{4a^2} + \frac{b^2}{4a^2}} [/tex] [tex]x=- \frac{b}{2a} +/-\sqrt{- \frac{4ac+b^2}{4a^2} } [/tex]