Question

Obtain the discriminant of the equation 2x^3-23x^2+82x-78=0
Hence provide the nature of its roots.

Answer 1

Answer on Question #78502 – Math – Other

Question

Obtain the discriminant of the equation $2x^{3} - 23x^{2} + 82x - 78 = 0$. Hence provide the nature of its roots.

Solution

The discriminant of a cubic equation $ax^3 + bx^2 + cx + d = 0$ is given by

$$\Delta_3 = b^2 c^2 - 4ac^3 - 4b^3 d - 27a^2 d^2 + 18abcd$$

In our case

$$\begin{array}{l} \Delta_3 = (-23)^2 82^2 - 4 \cdot 2 \cdot 82^3 - 4 \cdot (-23)^3 (-78) - 27(2)^2 (-78)^2 + 18 \cdot 2 \cdot (-23)82(-78) \\ = -11236 \end{array}$$

Since $\Delta_3 < 0$, the equation has one real root and two complex conjugate roots.

**Answer**: $-11236$, one real root and two complex conjugate roots.

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