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What are the possible solutions for the equation: \(\sqrt{2x - 1} - (2 - x) = 0\)?

a) \(x = 1\)
b) \(x = 2\)
c) \(x = 3\)
d) \(x = 4\)

Choose the correct possible solutions for the given equation.

User JSCard
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Final Answer:

The possible solutions for the equation
\(√(2x - 1) - (2 - x) = 0\) are \(x = 1\) and \(x = 2\). Therefore, the correct option is a) x = 1.

Step-by-step explanation:

To find the solutions of this equation, we need to isolate the square root and then square both sides.

First, we add (2 - x) to both sides:


\[√(2x - 1) = 2 - x\]

Now, we square both sides:


\[2x - 1 = (2 - x)^2\]

Expanding the right-hand side:


\[2x - 1 = 4 - 4x + x^2\]

Collecting terms with \(x^2\):


x^2 + (-4)x + (-3) = 0

This is a quadratic equation in the form of a
x^2 + bx + c = 0. We can find the roots using the quadratic formula:


\[x = (-b \pm √(b^2 - 4ac))/(2a)\]

Substituting our values for a, b, and c:


\[x = (4 \pm √(16 + 12))/(4)\]

Simplifying:


\[x = (1 \pm √(5))/(2)\]

We can't take the square root of a negative number, so we discard the negative value:


\[x = (1 + √(5))/(2)\]or
\[x = (1 - √(5))/(2)\] (but this is not a valid solution since it gives a negative value inside the square root in the original equation)

Now, we know that x is either 1.618... or 0.618..., but these decimal approximations are not exact solutions. We can simplify them by using the fact that
\(√(5) \approx 2.236... and rounding to two decimal places:


\[x \approx 1.62\text{ or }0.62\] (but again, only x=1 is a valid solution since it gives a non-negative value inside the square root in the original equation) Therefore, the correct option is a) x = 1.

User UWSkeletor
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