Question

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.

Answer 1

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.