Analyzing Quadratic Roots with Equal Magnitude and Opposite Sign

The problem asks us to find the value of \(k\) for which the quadratic equation \(3x^2 - (k^2 + 5k)x + 3k^2 - 5k = 0\) has real roots of equal magnitude and opposite sign.

Let the given quadratic equation be \(ax^2 + bx + c = 0\). In this equation:

• \(a = 3\)
• \(b = -(k^2 + 5k)\)
• \(c = 3k^2 - 5k\)

For a quadratic equation to have real roots of equal magnitude and opposite sign, two main conditions must be met:

1. The roots must be real. This means the discriminant, \(\Delta = b^2 - 4ac\), must be non-negative, i.e., \(\Delta \ge 0\).
2. The roots must be of equal magnitude and opposite sign. If the roots are \(\alpha\) and \(\beta\), this condition means \(\beta = -\alpha\). The sum of the roots, \(\alpha + \beta\), must be zero. For a quadratic equation, the sum of the roots is given by \(-b/a\). Therefore, we must have \(-b/a = 0\), which implies \(b = 0\).

Let's apply these conditions to the given equation.

Condition 1: Sum of Roots is Zero (b = 0)

The coefficient of the \(x\) term is \(b = -(k^2 + 5k)\). Setting this to zero gives:

\(-(k^2 + 5k) = 0\)

\(k^2 + 5k = 0\)

\(k(k + 5) = 0\)

This equation gives two possible values for \(k\):

• \(k = 0\)
• \(k + 5 = 0 \implies k = -5\)

These are the values of \(k\) for which the sum of roots is zero. Now, we must check if for these values of \(k\), the roots are also real.

Condition 2: Real Roots (\(\Delta \ge 0\))

The discriminant is \(\Delta = b^2 - 4ac\). Substituting the values of \(a, b, c\):

\(\Delta = (-(k^2 + 5k))^2 - 4(3)(3k^2 - 5k)\)

\(\Delta = (k^2 + 5k)^2 - 12(3k^2 - 5k)\)

\(\Delta = k^2(k+5)^2 - 36k^2 + 60k\)

\(\Delta = k^2(k^2 + 10k + 25) - 36k^2 + 60k\)

\(\Delta = k^4 + 10k^3 + 25k^2 - 36k^2 + 60k\)

\(\Delta = k^4 + 10k^3 - 11k^2 + 60k\)

\(\Delta = k(k^3 + 10k^2 - 11k + 60)\)

For real roots, we need \(\Delta \ge 0\).

Checking the Possible Values of k

We found two possible values for \(k\) from the sum of roots condition: \(k=0\) and \(k=-5\).

Case 1: \(k = 0\)

If \(k=0\), the quadratic equation becomes:

\(3x^2 - (0^2 + 5(0))x + 3(0)^2 - 5(0) = 0\)

\(3x^2 - 0x + 0 = 0\)

\(3x^2 = 0\)

\(x^2 = 0\)

The roots are \(x = 0, 0\). These are real roots and they have equal magnitude (0), but they are not of opposite sign (0 is not considered to have an opposite sign other than itself). Thus, \(k=0\) does not satisfy the condition of having roots of equal magnitude and opposite sign.

Let's check the discriminant for \(k=0\):

\(\Delta = 0((0)^3 + 10(0)^2 - 11(0) + 60) = 0(60) = 0\)

Since \(\Delta = 0\), the roots are real and equal, which confirms \(x=0, 0\). But they are not opposite in sign.

Case 2: \(k = -5\)

If \(k = -5\), the quadratic equation becomes:

\(3x^2 - ((-5)^2 + 5(-5))x + 3(-5)^2 - 5(-5) = 0\)

\(3x^2 - (25 - 25)x + 3(25) + 25 = 0\)

\(3x^2 - 0x + 75 + 25 = 0\)

\(3x^2 + 100 = 0\)

\(x^2 = -\frac{100}{3}\)

\(x = \pm \sqrt{-\frac{100}{3}} = \pm i \sqrt{\frac{100}{3}}\)

The roots are imaginary (\(\pm i \frac{10}{\sqrt{3}}\)). These are not real roots.

Let's check the discriminant for \(k = -5\):

\(\Delta = -5((-5)^3 + 10(-5)^2 - 11(-5) + 60)\)

\(\Delta = -5(-125 + 10(25) + 55 + 60)\)

\(\Delta = -5(-125 + 250 + 55 + 60)\)

\(\Delta = -5(125 + 55 + 60)\)

\(\Delta = -5(240) = -1200\)

Since \(\Delta = -1200 < 0\), the roots are imaginary, which confirms our finding above. Thus, \(k=-5\) does not satisfy the condition of having real roots.

Conclusion on Existence of k

We found that for the sum of the roots to be zero (a necessary condition for roots of equal magnitude and opposite sign), \(k\) must be \(0\) or \(-5\). However, for \(k=0\), the roots are real but not of opposite sign. For \(k=-5\), the roots are of opposite sign but not real.

Therefore, there is no value of \(k\) for which the quadratic equation \(3x^2 - (k^2 + 5k)x + 3k^2 - 5k = 0\) has real roots of equal magnitude and opposite sign.

Based on this analysis, the correct option is that no such value of \(k\) exists.

Revision Table: Quadratic Roots Properties

Additional Information on Quadratic Equation Roots

The nature of the roots of a quadratic equation \(ax^2 + bx + c = 0\) depends entirely on the value of the discriminant, \(\Delta = b^2 - 4ac\). This value tells us whether the roots are real or imaginary, and if real, whether they are distinct or repeated.

• If \(\Delta > 0\), the equation has two distinct real roots.
• If \(\Delta = 0\), the equation has exactly one real root (or two equal real roots).
• If \(\Delta < 0\), the equation has two complex conjugate roots (imaginary roots).

The sum and product of the roots are also crucial properties. For \(ax^2 + bx + c = 0\):

• Sum of roots (\(\alpha + \beta\)) = \(-b/a\)
• Product of roots (\(\alpha \beta\)) = \(c/a\)

In this specific problem, the condition "equal magnitude and opposite sign" for real roots means the roots are of the form \(\alpha\) and \(-\alpha\), where \(\alpha\) is a non-zero real number. The sum of these roots is \(\alpha + (-\alpha) = 0\). This directly implies that the coefficient \(b\) must be zero. We must also ensure the roots are real, which requires \(\Delta \ge 0\). Both conditions must be simultaneously satisfied for the same value of \(k\).