What is the sum of all real roots of the equation 3x^4 + 10x^2 - 8 = 0?

What is the sum of all real roots of the equation 3x^4 + 10x^2 - 8 = 0?

• A. -8
• B. 3
• C. 0
• D. 8

Answer: (C)

To find the sum of all real roots of the equation \( 3x^4 + 10x^2 - 8 = 0 \), we can simplify the problem by making a substitution. Let \( y = x^2 \). This transforms the original equation into a quadratic equation: \[ 3y^2 + 10y - 8 = 0 \] We can solve this quadratic equation using the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 3 \), \( b = 10 \), and \( c = -8 \). Calculating the discriminant: \[ b^2 - 4ac = 10^2 - 4(3)(-8) = 100 + 96 = 196 \] Since the discriminant is positive, there will be two distinct real roots for \( y \). Now we can find the roots: \[ y = \frac{-10 \pm \sqrt{196}}{2 \cdot 3} = \frac{-10 \pm 14}{6} \] Calculating the two possible values

To find the sum of all real roots of the equation ( 3x^4 + 10x^2 - 8 = 0 ), we can simplify the problem by making a substitution. Let ( y = x^2 ). This transforms the original equation into a quadratic equation:

[

3y^2 + 10y - 8 = 0

]

We can solve this quadratic equation using the quadratic formula:

[

y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

]

where ( a = 3 ), ( b = 10 ), and ( c = -8 ). Calculating the discriminant:

[

b^2 - 4ac = 10^2 - 4(3)(-8) = 100 + 96 = 196

]

Since the discriminant is positive, there will be two distinct real roots for ( y ). Now we can find the roots:

[

y = \frac{-10 \pm \sqrt{196}}{2 \cdot 3} = \frac{-10 \pm 14}{6}

]

Calculating the two possible values