Question

If \(\mathrm{x}>1\) then the square roots of \(1-\sqrt{\left(\mathrm{x}^{2}-1\right) \mathrm{i}}\) (a) \(\pm[\sqrt{\\{}(\mathrm{x}+1) / 2\\}-\mathrm{i} \sqrt{\\{}(\mathrm{x}-1) / 2\\}]\) (b) \(\pm[\sqrt{\\{}(\mathrm{x}+1) / 2\\}+\mathrm{i} \sqrt{\\{}(\mathrm{x}-1) / 2\\}]\) (c) \(\pm[\sqrt{\\{}(\mathrm{x}-1) / 2\\}-\mathrm{i} \sqrt{\\{}(\mathrm{x}+1) / 2\\}]\) (d) \(\pm[\sqrt{\\{}(\mathrm{x}-1) / 2\\}+\mathrm{i} \sqrt{\\{}(\mathrm{x}+1) / 2\\}]\)

Short answer

The short answer to the question is: (b) $\pm[\sqrt{(\mathrm{x}+1) / 2}-\mathrm{i} \sqrt{(\mathrm{x}-1) / 2}]\( or \)\pm[\sqrt{(\mathrm{x}+1) / 2}+\mathrm{i} \sqrt{(\mathrm{x}-1) / 2}]$

Step-by-step solution

Find the square root of the expression

To find the square root of the given expression \(1-\sqrt{(x^2-1)i}\), we can call it \(z\), and square both sides of the equation: \(z^2 = 1-\sqrt{(x^2-1)i}\)

Expand the square of z

To simplify the equation further, assume \(z = a+bi\), where \(a\) and \(b\) are the real and imaginary components of \(z\). By squaring the left side, we get \((a+bi)^2 = a^2 + 2abi - b^2 = 1-\sqrt{(x^2-1)i}\)

Separate real and imaginary components

Now, let's separate the real and imaginary parts in the equation: Real part: \(a^2 - b^2 = 1\) Imaginary part: \(2ab = -\sqrt{x^2-1}\)

Solve for a and b

From the real part equation, we have: \(a^2 = b^2+1\) Now take the square root from the imaginary part's equation in both sides: \(b = -\frac{\sqrt{x^2-1}}{2a}\) Square both sides of this equation: \(b^2 = \frac{x^2-1}{4a^2}\)

Substitute b expression

Now, substitute the expression for \(b^2\) into the equation for \(a^2\): \(a^2 = \frac{x^2-1}{4a^2} + 1\)

Solve for a

Multiply both sides by \(4a^2\): \(4a^4 = x^2-1 + 4a^2\) We want to find a and b separately. Rearrange the equation to get}, \(4a^4-4a^2-x^2+1=0\) Let \(k = a^2\). Therefore, the equation can be written as: \(4k^2 - 4k - x^2 + 1 = 0\) We can solve for k first, then find a.

Solve for k

Given the quadratic equation, \(k = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(4)(-x^2 + 1)}}{2(4)}\) Solving for k, \(k = \frac{4 \pm \sqrt{16 + 16(x^2 - 1)}}{8}\) \(k = \frac{1}{2} \pm \frac{1}{2}\sqrt{x^2 - 1}\) Now, to find a, \(a = \pm\sqrt{k} = \pm\sqrt{\frac{1}{2} \pm \frac{1}{2}\sqrt{x^2 - 1}}\)

Solve for b

Using the expression derived in step 4 for b, we substitute \(k = a^2\): \(b = -\frac{\sqrt{x^2-1}}{2\sqrt{k}}\) Substitute the expression for a, \(b = \pm\sqrt{\frac{1}{2} \mp \frac{1}{2}\sqrt{x^2 - 1}}\)

Compare with the given options

Comparing the derived expressions for a and b to the given options, we can match the result to option (b): $z=\pm[\sqrt{(\mathrm{x}+1) / 2}-\mathrm{i} \sqrt{(\mathrm{x}-1) / 2}]\( or \)\pm[\sqrt{(\mathrm{x}+1) / 2}+\mathrm{i} \sqrt{(\mathrm{x}-1) / 2}]$

Key concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics and engineering, representing numbers that have both a real part and an imaginary part. They are written in the form of \(a + bi\), where \(a\) is the real component and \(b\) is the imaginary component multiplied by \(i\), the imaginary unit where \(i^2 = -1\). Understanding complex numbers is crucial for solving a variety of problems, including those related to the square roots of negative numbers and solutions of quadratic equations that do not have real solutions.

The relevance of complex numbers can't be overstated, as they expand our mathematical horizons beyond the limitations of just the real numbers. For example, while the square root of a negative number is not defined in the real numbers, complex numbers provide a graceful and useful solution to this issue. This allows for the handling of equations and physical problems that would otherwise be unsolvable.

Square Root Calculation

The process of finding the square root of a number involves identifying a value that, when multiplied by itself, will yield the original number. Calculating square roots is straightforward for positive real numbers, but it introduces the concept of complex numbers when dealing with negative radicands (the number under the square root symbol).

In the case of finding the square root of a complex expression such as \(1-\sqrt{(x^2-1)i}\), we must apply algebraic manipulation and knowledge of complex numbers to simplify the expression. The square root calculation in the context of complex numbers often entails finding two components: a real part and an imaginary part, which correspond to \(a\) and \(bi\) of a complex number in the form \(a + bi\). This can involve completing the square, using properties of \(i\), and equating real and imaginary parts to solve for the components separately.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically in the form of \(ax^2 + bx + c = 0\). These equations are pivotal in various areas of mathematics and science. Solving quadratic equations can be performed using several techniques, including factoring, completing the square, using the quadratic formula, or graphing. The quadratic formula, in particular, \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), is an extremely powerful tool in solving these equations, especially when the discriminant \(b^2-4ac\) is negative, resulting in complex solutions.

In practice, as with the exercise provided, one may encounter a scenario where the solution to a quadratic equation involves finding a complex number that is itself squared. This requires transforming the quadratic equation into a new variable, such as \(k = a^2\), to determine the square of the real part of the complex solution. Once \(k\) has been found, the process of taking square roots returns us to the original variable and completes the solution to the problem. This method of solving through substitution is not only useful in finding the roots of the equation but also illuminates the interplay between complex numbers and quadratic solutions.