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`:

`(ax^2+bx+c)/(a) = (0)/(a)`

`(ax^2)/(a) + (bx)/(c) + (c)/(a) =0`

`x^(2) + (b)/(a) x+ (c)/(a) =0`

Step 3. Subtract
`(c)/(a)` from both sides of the equation:

`x^(2) + (b)/(a) x+ (c)/(a)- (c)/(a) =0 - (c)/(a)`

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

Step 4. Complete the square of the left hand side by adding
`( (b)/(2a) )^2` to both sides:

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

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

Step 5. Take square root to both sides of the equation:

`\sqrt{(x+ (b)/(2a) )^2} =+/- \sqrt{- (c)/(a) +( (b)/(2a) )^2}`

`x+ (b)/(2a)=+/-\sqrt{- (c)/(a) +( (b)/(2a) )^2}`

Step 6. Subtract
`(b)/(2a)` from both sides of the equation:

`x+ (b)/(2a)- (b)/(2a) =- (b)/(2a) +/-\sqrt{- (c)/(a) +( (b)/(2a) )^2}`

`x=- (b)/(2a) +/-\sqrt{- (c)/(a) +( (b)/(2a) )^2}`

Step 7. Simplify the radicand of the left hand side of the equation:

`x=- (b)/(2a) +/-\sqrt{- (c)/(a) + (b^2)/((2a)^2)}`

`x=- (b)/(2a) +/-\sqrt{- (c)/(a) + (b^2)/(4a^2)}`

`x=- (b)/(2a) +/-\sqrt{- (4ac)/(4a^2) + (b^2)/(4a^2)}`

`x=- (b)/(2a) +/-\sqrt{- (4ac+b^2)/(4a^2) }`

Step 8. Take
`4a^2` outside the radical:

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

`x=- (b)/(2a) +/- ( √(-4ac+b^2) )/(2a) }`

Step 9. Combine the two fractions and rearrange the terms in the radical:

`x= (-b+/- √(b^2-4ac) )/(2a)`