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anemone
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If the equation $ax^2+(c-b)x+e-d=0$ has real solutions that are greater than 1, prove that the equation $ax^4+bx^3+cx^2+dx+e=0$ has at least one real solution.
[sp]Let $s$ be a solution of $ax^2+(c-b)x+e-d=0$ with $s>1$. Then $as^2 + cs + e = bs+d.$anemone said:If the equation $ax^2+(c-b)x+e-d=0$ has real solutions that are greater than 1, prove that the equation $ax^4+bx^3+cx^2+dx+e=0$ has at least one real solution.
Opalg said:[sp]Let $s$ be a solution of $ax^2+(c-b)x+e-d=0$ with $s>1$. Then $as^2 + cs + e = bs+d.$
Let $f(x) = ax^4+bx^3+cx^2+dx+e$. Then $f(\sqrt s) = as^2 + cs + e + \sqrt s(bs+d) = (bs+d)(1 + \sqrt s).$ In the same way, $f(-\sqrt s) = (bs+d)(1- \sqrt s).$
If $bs+d = 0$ then $\sqrt s$ and $ - \sqrt s$ are zeros of $f(x)$. If $bs+d \ne 0$ then $f(\sqrt s)$ and $f(-\sqrt s)$ have opposite signs (because $\sqrt s > 1$). So by the intermediate value theorem $f(x)$ must take the value zero somewhere between $-\sqrt s$ and $\sqrt s$. In either case, the equation $ax^4+bx^3+cx^2+dx+e=0$ has at least one real solution.
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The most common method is to use the fundamental theorem of algebra, which states that a polynomial of degree n has n complex roots. Since a quartic equation has degree 4, it must have at least 1 real solution. This is because complex roots always come in pairs, so if there are 4 complex roots, there must be 2 real roots.
One example is the equation x4 + 3x3 + 2x2 + 4x + 3 = 0. This equation has one real solution, which is approximately -0.835.
Yes, another method is to use the Descartes' rule of signs. This rule states that the number of positive real roots of a polynomial is equal to the number of sign changes in the coefficients or less by an even number. Similarly, the number of negative real roots is equal to the number of sign changes in the coefficients or less by an odd number.
Yes, it is possible for a quartic equation to have up to 4 real solutions. However, it is also possible for it to have 2 or 3 real solutions. The number of real solutions depends on the coefficients of the equation.
Yes, a quartic equation can have no real solutions. This is because the fundamental theorem of algebra and Descartes' rule of signs only guarantee the existence of at least one real solution, but not necessarily more. In rare cases, all 4 roots of a quartic equation may be complex, meaning there are no real solutions.