Question

Begin with quadratic equation ax^2+bx+c=0 Solve for x using the completing the square method deriving the quadratic formula.

Answer 1

Consider the quadratic equation:

`ax^2+bx+c=0`

to solve this quadratic equation by completing the square, we can perform the following steps:

Step 1: transform the equation so that the constant term c is alone on the right side:

`ax^2+bx=-c`

Step 2: If a, the leading coefficient, is not equal to 1, then divide both sides of the above equation by a:

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

Step 3: add the square of half the coefficient of the x-term, to both sides of the equation:

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

Step 4: Factor the left side as the square of a binomial:

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

now, if we denote by q = b/(2a) and by r the right side of the above equation, we get:

`(x+q)^2=r`

Step 5: Take the square root of both sides, to obtain:

`x+q^{}=\pm\sqrt[]{r}`

Step 6: solve for x:

`x=\pm\sqrt[]{r}\text{ - q}`