Question 225322
First let me say that this is an odd (and incomplete) way of completing the square. Anyway, let's proceed...<br>
{{{4x^2-4x + 3 = 0}}}
a. move the constant term to the right side of the equation
{{{4x^2-4x = -3}}}
b. multiply each term in the equation by four times the cofficient of the x2 term
4*4 = 16 so:
{{{16(4x^2-4x) = 16(-3)}}}
{{{64x^2 - 64x = -48}}}
c. square the coefficient of the original x term and add it to both sides of the equation
{{{(-4)^2 = 16}}} so:
{{{64x^2 - 64x + 16 = -32}}}
A step that is missing here is to write the left side as a perfect square. After the steps so far the left side should fit one of the following patterns:
{{{a^2 +2ab + b^2 = (a+b)^2}}} or {{{a^2 -2ab + b^2 = (a-b)^2}}}. Your equation fits the second pattern with "8x" as "a" and "4" as "b":
{{{(8x - 4)^2 = -32}}}
d. take the square root of both sides
At this point, with this equation, we have a problem. The left side is a perfect square. The right side is a negative number. Unless we are working with complex numbers, there are no perfect squares that are negative. So this equation has no solutions.<br>
{{{x^2 + 12x -64 = 0}}}
a. move the constant term to the right side of the equation
{{{x^2 + 12x = 64}}}
b. multiply each term in the equation by four times the cofficient of the x2 term
4*1 = 4 so:
{{{4(x^2 + 12x) = 4(64)}}}
{{{4x^2 + 48x = 256}}}
c. square the coefficient of the original x term and add it to both sides of the equation
{{{12^2 = 144}}} so:
{{{4x^2 + 48x + 144 = 400}}}
Missing step: Rewrite as a perfect square. The left side fits the first perfect square pattern (see above) with "2x" as "a" and "12" as "b":
{{{(2x + 12)^2 = 400}}}
d. take the square root of both sides
{{{sqrt((2x + 12)^2) = sqrt(400)}}}
e. set the left side of the equation equal to the positive square root of the number on the right side and slove for x
Since {{{sqrt(400) = 20}}}:
2x + 12 = 20
2x = 8
x = 4
f. set the left side of the equation equal to the negative square root of the number on the right side of the equation and slove for x
2x + 12 = -20
2x = -32
x = -16