Find all real values of a for which the equation \((x^2 + a)^2 + a = x\) has four real roots.
+26367 Find all real values of a for which the equation \(\mathbf{(x^2 + a)^2 + a = x}\) has four real roots.
My attempt:
I assume there are 4 distinct real roots. So there are 3 local maxima/minima.
We find the local maxima/minima by differentiation. Maxima/minima occur when f'(x) = 0
1. Differentiation:
\(\begin{array}{|rcll|} \hline y &=& (x^2 + a)^2 -x+a \\ y' &=& 2(x^2+a)\cdot 2x-1 \\ 0 &=& 4x(x^2+a) - 1 \\ 4x(x^2+a) &=& 1 \quad | \quad : 4 \\ x(x^2+a) &=& \frac14 \\ \mathbf{x^3+ax-\frac14} & \mathbf{=} & \mathbf{0} \\ \hline \end{array}\)
The zero set of discriminant of the cubic \(\mathbf{Ax^{3}+Bx^{2}+Cx+D\,}\) has discriminant
\(\mathbf{ B^{2}C^{2}-4AC^{3}-4B^{3}D-27A^{2}D^{2}+18ABCD\,.}\)
The discriminant is zero if and only if at least two roots are equal.
If the coefficients are real numbers, and the discriminant is not zero,
the discriminant is positive if the roots are three distinct real numbers,
and negative if there is one real root and two complex conjugate roots.
\(\begin{array}{|rcll|} \hline \mathbf{Ax^{3}+Bx^{2}+Cx+D\,} \\ \mathbf{x^3+ax-\frac14} & \mathbf{=} & \mathbf{0} \quad | \quad A=1,\ B=0,\ C=a,\ D=-\frac14 \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \mathbf{ B^{2}C^{2}-4AC^{3}-4B^{3}D-27A^{2}D^{2}+18ABCD } &>& 0 \quad | \quad \text{there are three distinct real numbers} \\ -4a^3 -27\left(-\frac14\right)^2 &>& 0 \\ -4a^3 -\frac{27}{16} &>& 0 \\ -4a^3 &>& \frac{27}{16} \quad | \quad : -4 \\ a^3 &<& -\frac{27}{64} \\ a^3 &<& -\frac{3^3}{4^3} \\ \mathbf{ a } &\mathbf{<}& \mathbf{-\frac{3}{4}} \\ \hline \end{array}\)
\(\mathbf{(x^2 + a)^2 + a = x}\) has four distinct real roots if \(\mathbf{ a<-\frac{3}{4}}\)
+26367 Find all real values of a for which the equation \(\mathbf{(x^2 + a)^2 + a = x}\) has four real roots.
My attempt:
I assume there are 4 distinct real roots. So there are 3 local maxima/minima.
We find the local maxima/minima by differentiation. Maxima/minima occur when f'(x) = 0
1. Differentiation:
\(\begin{array}{|rcll|} \hline y &=& (x^2 + a)^2 -x+a \\ y' &=& 2(x^2+a)\cdot 2x-1 \\ 0 &=& 4x(x^2+a) - 1 \\ 4x(x^2+a) &=& 1 \quad | \quad : 4 \\ x(x^2+a) &=& \frac14 \\ \mathbf{x^3+ax-\frac14} & \mathbf{=} & \mathbf{0} \\ \hline \end{array}\)
The zero set of discriminant of the cubic \(\mathbf{Ax^{3}+Bx^{2}+Cx+D\,}\) has discriminant
\(\mathbf{ B^{2}C^{2}-4AC^{3}-4B^{3}D-27A^{2}D^{2}+18ABCD\,.}\)
The discriminant is zero if and only if at least two roots are equal.
If the coefficients are real numbers, and the discriminant is not zero,
the discriminant is positive if the roots are three distinct real numbers,
and negative if there is one real root and two complex conjugate roots.
\(\begin{array}{|rcll|} \hline \mathbf{Ax^{3}+Bx^{2}+Cx+D\,} \\ \mathbf{x^3+ax-\frac14} & \mathbf{=} & \mathbf{0} \quad | \quad A=1,\ B=0,\ C=a,\ D=-\frac14 \\ \hline \end{array}\)
\(\begin{array}{|rcll|} \hline \mathbf{ B^{2}C^{2}-4AC^{3}-4B^{3}D-27A^{2}D^{2}+18ABCD } &>& 0 \quad | \quad \text{there are three distinct real numbers} \\ -4a^3 -27\left(-\frac14\right)^2 &>& 0 \\ -4a^3 -\frac{27}{16} &>& 0 \\ -4a^3 &>& \frac{27}{16} \quad | \quad : -4 \\ a^3 &<& -\frac{27}{64} \\ a^3 &<& -\frac{3^3}{4^3} \\ \mathbf{ a } &\mathbf{<}& \mathbf{-\frac{3}{4}} \\ \hline \end{array}\)
\(\mathbf{(x^2 + a)^2 + a = x}\) has four distinct real roots if \(\mathbf{ a<-\frac{3}{4}}\)
