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anemone
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Let $a>b>c$ be the real roots of the equation $\sqrt{2014}x^3-4029x^2+2=0$. Find $b(a+c)$.
The formula for finding $b(a+c)$ in the equation $\sqrt{2014}x^3-4029x^2+2=0$ is $b(a+c) = \frac{-b}{a}$, where $a$ and $b$ are the coefficients of the quadratic term and the linear term, respectively.
A given equation has real roots if the discriminant, $b^2-4ac$, is greater than or equal to 0. If the discriminant is less than 0, the equation will have complex roots.
Yes, the value of $b(a+c)$ can be negative. This would occur if the coefficient of the linear term, $b$, is negative and the coefficient of the quadratic term, $a$, is positive.
Finding $b(a+c)$ for real roots is significant because it allows us to determine the sum of the roots of the equation. This can help us in solving the equation and understanding the behavior of the graph of the equation.
Yes, the value of $b(a+c)$ can be 0. This would occur if the coefficient of the linear term, $b$, is 0. In this case, the equation would simplify to $ax^2+c=0$ and the value of $b(a+c)$ would be 0 regardless of the value of $a$ and $c$.