Question

The equation \(x^{2}-3 k x+2 e^{\ln g_{x} k^{2}}-1=0\) has real roots such that the product of roots is 7 if the value of \(k\) is (a) 1 (b) 2 (c) 3 (d) 4

Short answer

Answer: (b) 2

Step-by-step solution

Find the product of roots from the given equation

The equation we are given is: \(x^2 - 3kx + 2e^{\ln g_{x} k^{2}} - 1 = 0\). For a quadratic equation of the form \(ax^2 + bx + c = 0\), the product of the roots is given by \(\frac{c}{a}\). So we have the product of roots as: \(\frac{2e^{\ln g_{x} k^{2}} - 1}{1}\).

Set the product of roots equal to 7 and solve for \(k\)

We know that the product of roots is 7, so we can set the expression from step 1 equal to 7 and solve for \(k\): \(\frac{2e^{\ln g_{x} k^{2}} - 1}{1} = 7\).

Simplify the expression and solve for \(k\)

First, eliminate the denominator of 1, so the equation becomes: \(2e^{\ln g_{x} k^{2}} - 1 = 7\). Next, add 1 to both sides of the equation: \(2e^{\ln g_{x} k^{2}} = 8\). Now, divide both sides of the equation by 2: \(e^{\ln g_{x} k^{2}} = 4\). Using properties of logarithms, we can simplify the equation further: \(g_{x} k^{2} = 4\). Since \(g_x\) is the base of logarithm, we can write the expression using logarithmic notation: \(\log_{g_x}(k^2) = \log_{g_x}(4)\). Since logarithms with the same base on both sides of the equation are equal, we can directly equate the arguments: \(k^2 = 4\). Now, taking the square root of both sides of the equation: \(k = \pm \sqrt{4}\). Finally, we get the value of \(k\): \(k = \pm 2\). Since the options given are 1, 2, 3 and 4, we choose the positive value of \(k\) matching one of the given options.

Find the correct answer from the given options

Now that we have found the correct value of \(k\), we can match it to the given options. We got \(k = 2\), which corresponds to: (b) 2. Therefore, the value of \(k\) for which the product of roots is 7 is 2.

Key concepts

Products of Roots

In a quadratic equation of the form \( ax^2 + bx + c = 0 \), the product of its roots is an essential concept. This product can be calculated using a straightforward formula: \( \frac{c}{a} \). This formula is derived from Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots.
When dealing with quadratic equations, identifying the product of roots is crucial in many applications, such as solving algebraic problems and analyzing roots' behavior. In our given equation \( x^2 - 3kx + 2e^{\ln g_x k^2} - 1 = 0 \), the product of the roots is \( \frac{2e^{\ln g_x k^2} - 1}{1} \).

• Set this product equal to 7, as specified in the problem.
• This allows us to find the specific value of \( k \) that satisfies this condition.

Understanding the product of roots helps in predicting how changing parameters affect the solutions to the equation, deepening comprehension of mathematical relationships.

Logarithmic Functions

Logarithms are the inverses of exponential functions. They allow us to solve equations where the unknown appears as an exponent. In the context of our problem, understanding logarithmic functions is essential because they appear within the exponent on the right-hand side of the equation.
Logarithmic identities are very helpful here. For example, the property \( e^{\ln x} = x \) demonstrates how exponential and logarithmic functions undo each other. In the equation \( e^{\ln g_x k^2} = 4 \), we use this identity to simplify and solve for \( k^2 \).

• This process involves recognizing that \( e^{\ln g_x k^2} \) simplifies directly to \( g_x k^2 \).
• Converting and simplifying logarithmic expressions can make complex algebraic manipulations much more accessible.

Understanding these properties aids in deciphering seemingly challenging exponential equations, such as finding the product of roots in this scenario.

Quadratic Formula

The quadratic formula provides a method to find the roots of any quadratic equation. Although not directly used in this problem's step-by-step solution, knowing this formula is beneficial. It is given as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula allows one to calculate roots directly, regardless of the factoring abilities of the quadratic expression. This can be particularly useful if simplifying or factoring seems daunting.
In our original equation, applying the quadratic formula could guide understanding the problem's dynamics more thoroughly. It relates the discriminant \( b^2 - 4ac \) to the nature of the roots. In other words:

• If the discriminant is positive, roots are real and distinct.
• If zero, roots are real and identical.
• If negative, roots are complex and not real.

Thus, although not employed step-by-step in this case, the quadratic formula helps in validating the realness of roots and providing a safety net when other methods may seem evasive. Having this in your mathematical toolbox enriches problem-solving capabilities.