Question

If the equation \((3 x)^{2}+\left(27 \times 3^{1 / k}-15\right) x+4=0\) has equal roots, then \(k=\) (a) \(-2\) (b) \(-\frac{1}{2}\) (c) (d) 0

Short answer

Question: The quadratic equation \((3x)^2+(27\times3^{1/k}-15)x+4=0\) has equal roots. Find the value of \(k\). Answer: (a) \(-2\).

Step-by-step solution

Identify the coefficients of the quadratic equation

The given equation is \((3x)^2+(27\times3^{1/k}-15)x+4=0\). Comparing it with the standard quadratic equation (\(ax^2+bx+c=0\)), we can see that the coefficients are: \(a = (3)^2 = 9\) \(b = 27\times3^{1/k}-15\) \(c = 4\)

Calculate the discriminant

The discriminant \(D\) is given by the formula \(D=b^2-4ac\). Using the identified coefficients, we have: \(D=(27\times3^{1/k}-15)^2 - 4\times9\times4\)

Set the discriminant equal to 0 and solve for \(k\)

Since the quadratic equation has equal roots, the discriminant \(D\) must be equal to 0. So, we have: \((27\times3^{1/k}-15)^2 - 4\times9\times4 = 0\) Solving for \(k\), we get: \((27\times3^{1/k}-15)^2 = 4\times9\times4\) Taking the square root of both sides, we get: \(27\times3^{1/k}-15 = \pm 12\) Now we have two cases: Case 1: \(27\times3^{1/k}-15 = 12\) \(27\times3^{1/k} = 27\) \(3^{1/k} = 1\) Since any number raised to the power of 0 equals 1, \(k\rightarrow \infty\). However, this is not an option in the given choices. Case 2: \(27\times3^{1/k}-15 = -12\) \(27\times3^{1/k} = 3\) \(3^{1/k} = 3^{-2}\) \(1/k = -2\) Multiplying both sides by \(k\), we get: \(k=-2\). Therefore, the correct answer is (a) \(-2\).

Key concepts

Discriminant

The discriminant is a crucial component of quadratic equations. It helps determine the nature of the roots for the equation. Given the standard form of a quadratic equation: \[ ax^2 + bx + c = 0 \]The discriminant, represented as \(D\), is calculated using the formula: \[ D = b^2 - 4ac \] The value of \(D\) tells us about the roots:

• 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 equal root.
• If \(D < 0\), the equation has two complex roots.

In the exercise, because the roots are equal, \(D\) must be zero. Solving \(D = 0\) becomes key to finding the specific value of \(k\) in the equation. This practice provides insight into the behavior of quadratic functions and their roots.

Equal Roots

Equal roots refer to a scenario in a quadratic equation where both solutions for \(x\) are identical. This occurs specifically when the discriminant of the equation is zero. The concept of equal roots simplifies many quadratic problems because it means the parabola represented by the quadratic equation touches the x-axis at exactly one point. The formula for determining the position of these roots is: \[ x = \frac{-b \pm \sqrt{D}}{2a} \]When \(D = 0\), the equation turns into: \[ x = \frac{-b}{2a} \]This single solution means that the vertex of the parabola is on the x-axis. In the exercise, after setting \(D = 0\), the equations simplify, leading to the situation where only one value of \(k\) yields equal roots. Recognizing equal roots is powerful because it automatically conditions the equation for a single solution.

Coefficient

Coefficients are the numerical components multiplied by the variables in an equation, here in the quadratic form \(ax^2 + bx + c = 0\). They are vital in calculations involving quadratic equations.

• \(a\) is the coefficient of the \(x^2\) term, indicating the parabola's curvature direction and width.
• \(b\) is the coefficient of the \(x\) term, influencing the parabola's slope and orientation.
• \(c\) is the constant term, shifting the graph up or down along the y-axis.

In the given equation \((3x)^2 + (27 \times 3^{1/k} - 15)x + 4 = 0\), the coefficients are identified as \(a = 9\), \(b = 27 \times 3^{1/k} - 15\), and \(c = 4\). Recognizing these helps to apply the discriminant formula. This understanding is essential to solve quadratic equations and interpret their graphs in many mathematical applications. Using coefficients correctly leads directly to determining the condition for equal roots and solving for \(k\).