Question

The values of \(\mathrm{x}^{2}\) satisfying the equation \(\mathrm{x}^{4}-5 \mathrm{x}^{2}+6=0\) are (a) real and distinct (b) both positive (c) imaginary (d) both (a) and (b)

Short answer

a) Both roots are real. b) Both roots have positive square. c) Both roots are imaginary. d) both (a) and (b) Answer: d) both (a) and (b)

Step-by-step solution

Substitution and Simplification

Let \(\mathrm{y} = \mathrm{x}^2\). Then, the equation becomes \(\mathrm{y}^2 - 5\mathrm{y} + 6=0\).

Solve the Quadratic Equation

To solve the quadratic equation, we first calculate the discriminant: \[D = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1\] Since the discriminant is positive, we have real and distinct roots for the quadratic equation in \(\mathrm{y}\): \[\mathrm{y}_1 = \frac{-b + \sqrt{D}}{2a}=\frac{5 + 1}{2} = 3\] \[\mathrm{y}_2 = \frac{-b - \sqrt{D}}{2a}=\frac{5 - 1}{2} = 2\]

Solve for \(\mathrm{x}^2\)

Now we need to find the corresponding \(\mathrm{x}\) values using our found \(\mathrm{y}\) values: \[\mathrm{x}^2 = \mathrm{y}_1 =3\] \[\mathrm{x}^2 = \mathrm{y}_2 =2\]

Determine the Nature of the Roots

We can see that both roots of the equation are real and distinct \(\mathrm{x}^2\) values and both are positive. Therefore, according to the given options, the correct answer is (d) both (a) and (b).

Key concepts

Real and Distinct Roots

A key concept in solving quadratic equations is determining whether the equation has real and distinct roots. When you solve a quadratic equation, the roots or solutions are the values of the variable that satisfy the equation. Roots can be either real or complex, and if they are real, they can either be distinct or identical. **Real and distinct roots** occur when the equation has two different real number solutions. These are found when the equation's discriminant is positive. In our example, we substituted variable \(\mathrm{y} = \mathrm{x}^2\) turning the original quartic equation into a quadratic form \(\mathrm{y}^2 - 5\mathrm{y} + 6 = 0\), and then solved it to reveal two distinct y-values. Thus, both roots are real and are separate, showing distinctness.

Positive Roots

Positive roots are solutions to an equation that are greater than zero. For many equations, knowing the positivity of the roots is important because it implies a certain behavior about the variable being solved.In our problem, after solving the quadratic equation in terms of \(\mathrm{y}\), we derived roots \(\mathrm{y}_1 = 3\) and \(\mathrm{y}_2 = 2\), leading to \(\mathrm{x}^2\) solutions of 3 and 2 respectively. Since both of these are positive numbers, this implies that upon further solving for \(\mathrm{x}\), we will be working with real and non-negative solutions. This establishes the positivity of the roots.

Discriminant in Quadratic Equations

The discriminant is a critical factor when working with quadratic equations as it tells us about the nature of the roots without actually solving the equation. It is part of the quadratic formula and is given by: \[ D = b^2 - 4ac \]Where \(a\), \(b\), and \(c\) are coefficients of the quadratic equation \(ax^2 + bx + c = 0\). By calculating the discriminant, we can predict:

• If \(D > 0\), the equation has real and distinct roots.
• If \(D = 0\), the equation has real and identical or repeated roots.
• If \(D < 0\), the equation has complex or imaginary roots.

In our example, the discriminant for the quadratic form \(\mathrm{y}^2 - 5\mathrm{y} + 6 = 0\) was calculated as 1, which is positive. Hence, this confirms the roots are real and distinct. The importance of understanding this lies in predicting and determining the usability of the roots in real-world contexts.