Multiple Choice Question (MCQ)

The condition for the polynomial equation of degree 4 ax^4+bx^2+c=0 to have four distinct real roots is

  1. ×

    b^2-4ac>0

  2. ×

    b^2-4ac<0

  3. ×

    b<0 and b^2-4ac>0

  4. b<0, b^2-4ac>0 and ac>0

Collected in the board: Quartic Equations

Steven Zheng posted 1 week ago

Answer

  1. The quartic equation ax^4+bx^2+c=0 can be taken as a quadratic equation in terms of x^2, of which the roots can be determined by the root formula

    x^2= \dfrac{-b\pm\sqrt{b^2-4ac} }{2a}

    In order for x^2 to have real roots, the discriminant must be great than 0, then

    b^2-4ac>0

    In order for x to have 4 distinct real roots, x^2 must be a positive real number, then we get

    \dfrac{-b\pm\sqrt{b^2-4ac} }{2a}>0

    Then we get

    b< 0 \;\text{and}\; ac >0

    So choice D is the correct.

    PS: it's also possible for the equation has all real roots when b^2-4ac=0 and b<0 (a>0), then the equation has 2 pair of equal real roots (not distinct 4 real roots).




Steven Zheng posted 1 week ago

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