Question

How do you solve the quadratic equation by completing the square: `4x^2 + 9 = 12x`?

Answer 1

`x=3/2` is a "double root" (a root of multiplicity 2 ).

Explanation:

First, rearrange so the `x^2` and `x` terms are on the left and the constant term is on the right: `4x^2-12x=-9`.

Next, factor out the coefficient of `x^2` on the left: `4(x^2-3x)=-9`.

Next, you can either divide both sides by 4 before completing the square or complete the square first. Let's complete the square on `x^2-3x` first. Take the coefficient of `x`, which is `-3`, divide it by 2 to get `-3/2`, then square this number to get `9/4`. Add that number inside the parentheses and balance on the right side of the equation to get:

`4(x^2-3x+9/4)=-9+4*9/4=0` (it doesn't usually come out to be zero on the right here, that's just a coincidence).

Once you've done this procedure, the expression inside the parentheses will be a perfect square. In this case, we can write:

`4(x-3/2)^2=0`

In general, we can now divide both sides by the coefficient in front, which is 4 in this case. Since `0/4=0`, we get `(x-3/2)^2=0`.

Now take the `\pm` square root of both sides. Since `\pm sqrt(0)=0`, there is only one equation and one root of the original equation: `x-3/2=0` so `x=3/2` is a root of multiplicity 2.