How do you solve the quadratic equation by completing the square: #x^2-4x+2=0#?

Question

Isaiah Anthony · Accepted Answer

Complete the square to find #(x-2)^2 = x^2-4x+4 = 2#

Hence #x = 2 +-sqrt(2)# Add #2# to both sides to get: #2 = x^2-4x+4 = (x-2)^2# So #x-2 = +-sqrt(2)# Add #2# to both sides to get: #x = 2 +-sqrt(2)# In most cases: #ax^2+bx+c = a(x+b/(2a))^2 + (c - b^2/(4a))# from which we can derive the quadratic formula for solutions of #ax^2+bx+c = 0#: #x = (-b +-sqrt(b^2-4ac))/(2a)# Notice the #b/(2a)# term that gives us: #a(x+b/(2a))^2 = ax^2+bx+b^2/(4a)#

Hazel Anglin · Answer

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

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

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

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

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

5. Solve for x:
   x = 2 ± √2

Therefore, the solutions to the quadratic equation x^2 - 4x + 2 = 0 are x = 2 + √2 and x = 2 - √2.

Jace Anderson · Answer

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

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

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

3. Simplify both sides of the equation:
$$ (x - 2)^2 = 2 $$

4. Take the square root of both sides, considering both the positive and negative square roots:
$$ x - 2 = \pm \sqrt{2} $$

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

Therefore, the solutions to the quadratic equation $x^2 - 4x + 2 = 0$ by completing the square are $x = 2 + \sqrt{2}$ and $x = 2 - \sqrt{2}$.