Question

The roots of the equation \(\mathrm{k}^{4}-7 \mathrm{k}^{2}-8=0\) are (a) \((\pm \mathrm{i}, \pm \sqrt{2})\) (b) \((\pm 2, \pm 3)\) (c) \((\pm i, \pm 2 \sqrt{2})\) (d) \((\pm 1, \pm 2)\)

Short answer

Answer: The roots are $\pm i$ and $\pm 2\sqrt{2}$.

Step-by-step solution

Rewrite the equation in terms of u=k^2

Let's set u = k^2. Then, the given equation becomes: u^2 - 7u - 8 = 0.

Find the roots of the quadratic equation in u

Now, we will use the quadratic formula to find the roots of this equation. The quadratic formula for an equation au^2 + bu + c = 0 is: $$ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. $$ Here, a=1, b=-7, and c=-8. Plugging these values into the formula, we get: $$ u = \frac{7 \pm \sqrt{(-7)^2 - 4(1)(-8)}}{2(1)} = \frac{7 \pm \sqrt{49+32}}{2} = \frac{7 \pm \sqrt{81}}{2} = \frac{7 \pm 9}{2}. $$ This gives us two possible values for u: 8 and -1.

Find the roots of the original equation in k

Now that we have the values of u, we can find the values of k. Recall that u = k^2, so we need to find the square roots of our values for u. For u=8: $$ k=\pm \sqrt{8} = \pm 2\sqrt{2}. $$ For u=-1: $$ k=\pm \sqrt{-1} = \pm i. $$ Therefore, the roots of the given equation are: $$ k = \pm i, \pm 2\sqrt{2}. $$ Thus the correct answer is option (c).

Key concepts

Algebraic Equations

Algebraic equations form the foundation of solving mathematical problems that involve unknown values. Generally, they consist of two sides linked by an equals sign, demonstrating that each side has the same value. For example, in a simple linear equation like \(2x + 3 = 7\), we are tasked with determining the value of \(x\) that makes the equation true.

Approaching the more complex quadratic equations, which involve terms up to \(x^2\), we employ a systematic strategy to find solutions. One way to tackle higher-power equations, such as the quartic equation \(k^4 - 7k^2 - 8 = 0\) in our exercise, is to simplify them into quadratic form by utilizing a substitution method. This example demonstrates a clever strategy: By recognizing that the equation resembles a quadratic in form, we can simplify it by letting \(u = k^2\), thus transforming the quartic equation into a more manageable quadratic one that can then be solved with familiar methods.

Complex Numbers

Complex numbers are an extension of the real numbers we are commonly familiar with in everyday mathematics. They include a real part and an imaginary part, represented as \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the imaginary unit satisfying the equation \(i^2 = -1\).

In many quadratic equations, when the discriminant — the part under the square root in the quadratic formula — is negative, we encounter a situation where no real solution exists. This conundrum led mathematicians to devise complex numbers as solutions to such equations. In our original problem, the root for \(u = -1\) cannot be resolved within the real number system, as it involves taking the square root of a negative number. Thus, it calls for the use of complex numbers, resulting in \(k = \text{\pm}i\), with \(i\) representing the square root of -1.

Quadratic Formulas

The quadratic formula is a powerful and ubiquitous tool in algebra used to solve quadratic equations of the form \(ax^2 + bx + c = 0\). The elegant formula is expressed as \(x = \frac{-b \text{\pm} \sqrt{b^2 - 4ac}}{2a}\). It systematically solves for the values of \(x\) that satisfy the equation, based on the coefficients \(a\), \(b\), and \(c\).

Using the quadratic formula involves identifying these coefficients and then performing calculations to arrive at the solutions, which could be either real or complex numbers. The term under the square root, known as the discriminant, determines the nature of the roots. If positive, there are two distinct real solutions; if zero, there's exactly one real solution; and if negative, we find two complex solutions, as seen in the step-by-step solution for our example. This formula is a cornerstone in algebra for its robustness and reliability in tackling a wide range of equations.