Question

For the equation \(\mathrm{x}^{2}+\mathrm{k}^{2}=(2 \mathrm{k}+2) \mathrm{x}, \mathrm{k} \in \mathrm{R}\) roots are complex then (a) \(\overline{k=\\{(-1) / 2\\}}\) (b) \(k>\\{(-1) / 2\\}\) (c) \(k>\\{(-1) / 2\\}\) (d) \(\\{(-1) / 2\\}<\mathrm{k}<0\)

Short answer

The correct option for the given condition, where the roots of the equation \(x^2 + k^2 = (2k + 2)x\) are complex, is (a) \(\overline{k=\{(-1) / 2\}}\).

Step-by-step solution

Rewrite the equation in standard quadratic form

The given equation is \(x^2 + k^2 = (2k + 2)x\). We can rewrite it in the standard quadratic form by moving all terms to the left side of the equation: \(x^2 - (2k + 2)x + k^2 = 0\)

Find the discriminant

The discriminant of a quadratic equation of the form \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). In our case, \(a = 1\), \(b = -(2k + 2)\), and \(c = k^2\). Plugging these values into the formula, we get: \(D = (-(2k + 2))^2 - 4(1)(k^2)\)

Simplify the discriminant

Let's simplify the expression for the discriminant: \(D = (-2k - 2)^2 - 4k^2\) Now expand and simplify the discriminant: \(D = 4k^2 + 8k + 4 - 4k^2 = 8k + 4\)

Determine the condition for complex roots

For the roots of the quadratic equation to be complex, the discriminant must be negative, i.e., \(D < 0\). So, we have: \(8k + 4 < 0\) Now, solve the inequality: \(k < -\dfrac{1}{2}\)

Match the inequality with given options

The inequality we found in step 4 is \(k < -\dfrac{1}{2}\). Comparing this with the given options, we can see that it matches option (a): (a) \(\overline{k=\{(-1) / 2\}}\) So, the correct answer is (a).

Key concepts

Discriminant of a Quadratic

In understanding the nature of the roots of quadratic equations, the discriminant plays a pivotal role. It is calculated from the standard form of a quadratic equation, which is given by \(ax^2 + bx + c = 0\). The discriminant, represented by \(D\), is obtained using the formula \(D = b^2 - 4ac\).

What does the discriminant tell us? If \(D > 0\), the equation has two distinct real roots. If \(D = 0\), the equation has exactly one real root, also known as a repeated or double root. Lastly, if \(D < 0\), the roots are complex and conjugate to each other, meaning they have the form \(a + bi\) and \(a - bi\), where \(i\) is the imaginary unit.

To see the discriminant in action, consider the earlier exercise with the equation \(x^2 + k^2 = (2k + 2)x\). After calculating the discriminant as \(8k + 4\), it was determined that for the roots to be complex, \(8k + 4 < 0\), leading to the conclusion that \(k < -\frac{1}{2}\).

Nature of Roots

The discriminant not only determines the quantity of roots but also their nature. For complex roots, the quadratic equation's discriminant will be negative and there won't be any real solution on the number line. These complex roots are always in pairs, and they are essential in various areas of physics and engineering where real solutions don't suffice.

In the context of the given exercise, knowing that the required roots are complex immediately informs us that the discriminant will be negative. This provides a clear direction when analyzing the equation and comparing options to identify the correct range of \(k\) values.

Solving Quadratic Inequalities

Solving quadratic inequalities like \(D < 0\) requires an understanding of number properties and inequality manipulation. The process often involves finding the critical values where the inequality expression equals zero and then testing intervals to determine where the inequality holds true.

In the solution provided, once the discriminant was simplified to \(8k + 4\), the next step was to identify when this expression is less than zero. The inequality was thus solved to identify that the range for \(k\) to satisfy the quadratic inequality was \(k < -\frac{1}{2}\), leading to the selection of the correct option. This step demonstrates one of the methods to handle inequalities arising in quadratic equations.

Quadratic Equations in Standard Form

For quadratic equations to be analyzed and solved systematically, they must first be written in the standard form, \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients and constants. From this form, it's easier to identify the coefficients needed to calculate the discriminant and to apply various methods of solving the quadratic equation, such as factoring, completing the square, or using the quadratic formula.

The exercise presented required rearranging the original equation into the standard form. Once this was achieved, the rest of the problem-solving process, including finding the discriminant and understanding the nature of the roots, followed suit. Ensuring that the equation is in the standard form is a fundamental step in solving any quadratic equation.