How do you solve using the completing the square method #x^2-4x+1=0#?

Question

Eva Abramson · Accepted Answer

See explanation...

In addition to completing the square, I will use the difference of squares identity, which can be written: #a^2-b^2 = (a-b)(a+b)# with #a=(x-2)# and #b=sqrt(3)# as follows: #0 = x^2-4x+1# #= x^2-4x+4-3# #=(x-2)^2-(sqrt(3))^2# #=((x-2)-sqrt(3))((x-2)+sqrt(3))# #=(x-2-sqrt(3))(x-2+sqrt(3))# So: #x=2+-sqrt(3)#

Savannah Anderson · Answer

undefined To solve the equation using the completing the square method, follow these steps:

1. Move the constant term to the other side of the equation:
   x^2 - 4x = -1

2. Take half of the coefficient of x, square it, and add it to both sides of the equation:
   x^2 - 4x + (-4/2)^2 = -1 + (-4/2)^2
   x^2 - 4x + 4 = -1 + 4

3. Simplify both sides:
   x^2 - 4x + 4 = 3

4. Rewrite the left side as a perfect square:
   (x - 2)^2 = 3

5. Take the square root of both sides:
   x - 2 = ±√3

6. Solve for x:
   x = 2 ± √3

So the solutions are x = 2 + √3 and x = 2 - √3.

Julia Adams · Answer

undefined To solve the quadratic equation $x^2 - 4x + 1 = 0$ using the completing the square method, follow these steps:

1. Move the constant term to the other side of the equation:
$$x^2 - 4x = -1$$

2. To complete the square, take half of the coefficient of $x$, square it, and add it to both sides of the equation:
$$x^2 - 4x + (-4/2)^2 = -1 + (-4/2)^2$$
$$x^2 - 4x + 4 = -1 + 4$$

3. Simplify both sides of the equation:
$$x^2 - 4x + 4 = 3$$

4. Rewrite the left side of the equation as a perfect square:
$$(x - 2)^2 = 3$$

5. Take the square root of both sides of the equation:
$$x - 2 = \pm \sqrt{3}$$

6. Solve for $x$ by adding 2 to both sides:
$$x = 2 \pm \sqrt{3}$$

So the solutions to the equation $x^2 - 4x + 1 = 0$ using the completing the square method are $x = 2 + \sqrt{3}$ and $x = 2 - \sqrt{3}$.