How many real and imaginary roots does the equation y2=x3-4x have
A)2 real, 1 imaginary
B)all real
C)all imaginary
D)2 imaginary, 1 real

Question

Quizplus · Accepted Answer

The given equation is $y^2 = x^3 - 4x$. This is a cubic equation in terms of $x$, and cubic equations always have at least one real root because the function must cross the x-axis at least once. The equation $y^2 = x^3 - 4x$ can be rewritten in terms of $x$ to find the x-intercepts (roots) of the graph. The roots of the equation are the values of $x$ for which $y=0$, so we solve $0 = x^3 - 4x = x(x^2 - 4) = x(x-2)(x+2)$. This gives us three real roots: $x = 0$, $x = 2$, and $x = -2$, indicating that all roots are real. There are no imaginary roots because the equation does not involve complex numbers or operations that would produce imaginary results.

How Many Real and Imaginary Roots Does the Equation Y2=x3-4x