Use completing the square to determine which of the following gives the solutions to 3x2 − 24x + 40 = 28.

Question

A. 2±4sqrt3
B. 4±2sqrt3
C. 3±3sqrt2
D. 6±2sqrt5

Answer 1

First we need to put the equation in the form of $ax^2 + bx + c = 0$ by moving the 28 to the other side:

$3x^2 - 24x + 40 - 28 = 0$

$3x^2 - 24x + 12 = 0$

Next, we complete the square by finding the value that when squared, gives us the constant term at the end. In this case, it would be $(24/2)^2 = 144$.

$3x^2 - 24x + 144 = 132$

$(3x - 12)^2 = 132$

Now we can rewrite the equation to find the solutions:

$(3x - 12)^2 = 132$

$3x - 12 = ±√132$

$3x - 12 = ±2√33$

$3x = 12 ± 2√33$

$x = 4 ± 2√33$

Therefore, the solutions to the equation are:

B. 4 ± 2√33

Answer 2

or
starting here
3x^2 - 24x + 12 = 0
divide by 3
x^2 - 8 x + 4 = 0
x^2 - 8 x = -4
x^2 - 8 x + 16 = -4 + 16
( x - 4 )^2 = 12
x - 4 = +/- sqrt 12
x = 4 +/- sqrt 12
= 4 +/- 2 sqrt 3

Answer 3

You are correct, we can also use the completing the square method in a different way:

Starting from $3x^2 - 24x + 12 = 0$, we divide by 3 to simplify:

$x^2 - 8x + 4 = 0$

Now we want to re-write this equation by completing the square:

$x^2 - 8x = -4$

$x^2 - 8x + 16 = -4 + 16$

$(x - 4)^2 = 12$

$x - 4 = ±\sqrt{12}$

$x = 4 ± 2\sqrt{3}$

Therefore, the solutions to the given equation are also $4 ± 2\sqrt{3}$.

So, the correct answer in this case would be:

B. 4 ± 2√3