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

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1 Answer
Mar 7, 2018

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)