give exact and approximate solutions to three decimal places
(x-2)^2=12
(x-2)^2 = 12.
Take sqrt of both sides:
x - 2 = +-2sqrt3,
X = 2 +- 2sqrt3,
X = 5.4641, and -1.4641.
To find the exact and approximate solutions to the equation (x-2)^2 = 12, follow these steps:
1. Expand the equation: (x-2)(x-2) = 12.
2. Distribute the terms: x^2 - 4x + 4 = 12.
3. Move the constant term to the right side of the equation: x^2 - 4x - 8 = 0.
4. Notice that we have a quadratic equation in the form of ax^2 + bx + c = 0, where a = 1, b = -4 and c = -8.
5. Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
Using the formula, substitute a = 1, b = -4, and c = -8 into the quadratic formula:
x = (-(-4) ± √((-4)^2 - 4(1)(-8))) / (2(1))
Simplifying further:
x = (4 ± √(16 + 32)) / 2
x = (4 ± √48) / 2
x = (4 ± 4√3) / 2
x = 2 ± 2√3
Exact solutions:
x = 2 + 2√3
x = 2 - 2√3
Approximate solutions (rounded to three decimal places):
x ≈ 5.464
x ≈ -1.464
Therefore, the exact solutions to the equation (x-2)^2 = 12 are x = 2 + 2√3 and x = 2 - 2√3. The approximate solutions are x ≈ 5.464 and x ≈ -1.464.