Question

Solve quadratic equation by completing the square. Express answer as exact roots ?

Answer 1

`x=-4+-2/3sqrt(33)`

Explanation:

The steps I will use are:

• Premultiply by `12` to make the leading term a perfect square and all of the coefficients of the quadratic into integers.

• Complete the square by expressing in the form `(ax)^2+2(ax)(b)+(b)^2-c`.

• Use the difference of squares identity to factor `A^2-B^2 = (A-B)(A+B)`.

• Simplify and deduce roots.

So:

`0 = 12(3/4x^2+6x+1)`

`color(white)(0) = 9x^2+72x+12`

`color(white)(0) = (3x)^2+2(3x)(12)+(12)^2-132`

`color(white)(0) = (3x+12)^2-(2sqrt(33))^2`

`color(white)(0) = ((3x+12)-2sqrt(33))((3x+12)+2sqrt(33))`

`color(white)(0) = (3x+12-2sqrt(33))(3x+12+2sqrt(33))`

Hence:

`3x = -12+-sqrt(33)`

So:

`x = 1/3(-12+-2sqrt(33)) = -4+-2/3sqrt(33)`