1. **State the problem:** We have a quadratic equation depending on a real parameter $k$: $$ (1+k^2)x^2 + 10kx - 6(9k^2 + 1) = 0. $$ We define $f(k)$ as the larger root of this quadratic. We want to find the minimum value $a$ of $f(k)$ over all real $k$, the maximum value $b$ of $f(k)$, and then compute $100a + b$. 2. **Recall the quadratic formula:** For $ax^2 + bx + c = 0$, roots are $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. $$ The larger root corresponds to the $+$ sign. 3. **Identify coefficients:** $$ a = 1 + k^2, \quad b = 10k, \quad c = -6(9k^2 + 1) = -54k^2 - 6. $$ 4. **Write the larger root $f(k)$:** $$ f(k) = \frac{-10k + \sqrt{(10k)^2 - 4(1+k^2)(-54k^2 - 6)}}{2(1+k^2)}. $$ 5. **Simplify the discriminant:** $$ \Delta = 100k^2 - 4(1+k^2)(-54k^2 - 6) = 100k^2 + 4(1+k^2)(54k^2 + 6). $$ Expand: $$ 4(1+k^2)(54k^2 + 6) = 4(54k^2 + 6 + 54k^4 + 6k^2) = 4(54k^4 + 106k^2 + 6). $$ Calculate: $$ 4 \times 54k^4 = 216k^4, \quad 4 \times 106k^2 = 424k^2, \quad 4 \times 6 = 24. $$ So $$ \Delta = 100k^2 + 216k^4 + 424k^2 + 24 = 216k^4 + (100 + 424)k^2 + 24 = 216k^4 + 524k^2 + 24. $$ 6. **Rewrite $f(k)$:** $$ f(k) = \frac{-10k + \sqrt{216k^4 + 524k^2 + 24}}{2(1+k^2)}. $$ 7. **Analyze $f(k)$ to find extrema:** We want to find minimum and maximum values of $f(k)$ over $k \in \mathbb{R}$. 8. **Check behavior at infinity:** As $k \to \pm \infty$, leading terms dominate: $$ f(k) \approx \frac{-10k + \sqrt{216k^4}}{2k^2} = \frac{-10k + 14.6969 k^2}{2k^2} = \frac{-10k}{2k^2} + \frac{14.6969 k^2}{2k^2} = -\frac{5}{k} + 7.3485. $$ As $k \to \pm \infty$, $-5/k \to 0$, so $$ \lim_{k \to \pm \infty} f(k) = 7.3485. $$ 9. **Check $f(0)$:** $$ f(0) = \frac{0 + \sqrt{24}}{2} = \frac{4.898979}{2} = 2.4495. $$ 10. **Find critical points by differentiating $f(k)$:** Let $$ f(k) = \frac{-10k + \sqrt{216k^4 + 524k^2 + 24}}{2(1+k^2)}. $$ Set derivative $f'(k) = 0$ to find extrema. 11. **Set $t = k^2 \geq 0$ to simplify:** Rewrite $f(k)$ as function of $t$: $$ f(k) = \frac{-10k + \sqrt{216t^2 + 524t + 24}}{2(1+t)}. $$ Since $f(k)$ depends on $k$ and $-10k$ is odd in $k$, the function is not even in $k$. 12. **Check $k=0$ is not an extremum for $f(k)$ because derivative w.r.t. $k$ is not zero there (due to $-10k$ term).** 13. **Try to find $k$ such that numerator derivative equals zero:** Set derivative of numerator equal to zero: $$ -10 + \frac{1}{2\sqrt{216k^4 + 524k^2 + 24}} \times (864k^3 + 1048k) = 0. $$ Multiply both sides by $2\sqrt{216k^4 + 524k^2 + 24}$: $$ -20 \sqrt{216k^4 + 524k^2 + 24} + 864k^3 + 1048k = 0. $$ Rearranged: $$ 864k^3 + 1048k = 20 \sqrt{216k^4 + 524k^2 + 24}. $$ Square both sides: $$ (864k^3 + 1048k)^2 = 400 (216k^4 + 524k^2 + 24). $$ 14. **Let $t = k^2$, then $k^3 = k t$, so left side becomes:** $$ (864 k^3 + 1048 k)^2 = (k (864 t + 1048))^2 = k^2 (864 t + 1048)^2 = t (864 t + 1048)^2. $$ Right side: $$ 400 (216 t^2 + 524 t + 24). $$ 15. **Equation in $t$:** $$ t (864 t + 1048)^2 = 400 (216 t^2 + 524 t + 24). $$ Expand left side: $$ t (864^2 t^2 + 2 \times 864 \times 1048 t + 1048^2) = 400 (216 t^2 + 524 t + 24). $$ Calculate constants: $$ 864^2 = 746496, \quad 2 \times 864 \times 1048 = 1811712, \quad 1048^2 = 1098304. $$ So left side: $$ t (746496 t^2 + 1811712 t + 1098304) = 746496 t^3 + 1811712 t^2 + 1098304 t. $$ Right side: $$ 400 \times 216 t^2 + 400 \times 524 t + 400 \times 24 = 86400 t^2 + 209600 t + 9600. $$ 16. **Bring all terms to one side:** $$ 746496 t^3 + 1811712 t^2 + 1098304 t - 86400 t^2 - 209600 t - 9600 = 0, $$ which simplifies to $$ 746496 t^3 + (1811712 - 86400) t^2 + (1098304 - 209600) t - 9600 = 0, $$ $$ 746496 t^3 + 1725312 t^2 + 888704 t - 9600 = 0. $$ 17. **Divide entire equation by 64 to simplify:** $$ 11664 t^3 + 26958 t^2 + 13886 t - 150 = 0. $$ 18. **Solve cubic approximately:** Try $t=0.01$: $$ 11664 (0.000001) + 26958 (0.0001) + 13886 (0.01) - 150 \approx 0.0117 + 2.6958 + 138.86 - 150 = -8.43 < 0. $$ Try $t=0.02$: $$ 11664 (0.000008) + 26958 (0.0004) + 13886 (0.02) - 150 \approx 0.093 + 10.78 + 277.72 - 150 = 138.6 > 0. $$ Root between 0.01 and 0.02. Try $t=0.015$: $$ 11664 (0.000003375) + 26958 (0.000225) + 13886 (0.015) - 150 \approx 0.039 + 6.07 + 208.29 - 150 = 64.4 > 0. $$ Try $t=0.012$: $$ 11664 (0.000001728) + 26958 (0.000144) + 13886 (0.012) - 150 \approx 0.020 + 3.88 + 166.63 - 150 = 20.53 > 0. $$ Try $t=0.011$: $$ 11664 (0.000001331) + 26958 (0.000121) + 13886 (0.011) - 150 \approx 0.0155 + 3.26 + 152.75 - 150 = 6.02 > 0. $$ Try $t=0.0105$: $$ 11664 (0.000001157) + 26958 (0.000110) + 13886 (0.0105) - 150 \approx 0.0135 + 2.97 + 145.8 - 150 = -1.22 < 0. $$ So root near $t \approx 0.0107$. 19. **Calculate $k = \pm \sqrt{t} \approx \pm 0.103$.** 20. **Evaluate $f(k)$ at $k=0.103$:** Calculate numerator: $$ -10(0.103) + \sqrt{216 (0.103)^4 + 524 (0.103)^2 + 24}. $$ Calculate powers: $$ (0.103)^2 = 0.0107, \quad (0.103)^4 = (0.0107)^2 = 0.000114. $$ Calculate inside sqrt: $$ 216 \times 0.000114 + 524 \times 0.0107 + 24 = 0.0246 + 5.61 + 24 = 29.63. $$ Square root: $$ \sqrt{29.63} = 5.44. $$ Numerator: $$ -1.03 + 5.44 = 4.41. $$ Denominator: $$ 2(1 + 0.0107) = 2.0214. $$ So $$ f(0.103) = \frac{4.41}{2.0214} = 2.18. $$ 21. **Evaluate $f(k)$ at $k=-0.103$:** Numerator: $$ -10(-0.103) + 5.44 = 1.03 + 5.44 = 6.47. $$ Denominator same: 2.0214. $$ f(-0.103) = \frac{6.47}{2.0214} = 3.20. $$ 22. **Check values at $k=0$ and at infinity:** At $k=0$, $f(0) = 2.4495$. At $k=\pm \infty$, $f(k) \to 7.3485$. 23. **Summary of values:** - Minimum candidate near $k=0.103$ is $2.18$. - At $k=0$, $2.45$. - At $k=-0.103$, $3.20$. - At infinity, $7.35$. 24. **Check if $f(k)$ can be less than 2.18:** At $k=0$, $2.45 > 2.18$. At $k=0.103$, $2.18$ is minimum. 25. **Maximum is at infinity:** $$ b = \lim_{k \to \pm \infty} f(k) = 7.3485. $$ 26. **Calculate $100a + b$:** $$ 100 \times 2.18 + 7.3485 = 218 + 7.3485 = 225.3485. $$ **Final answer:** $$ \boxed{225.3485}. $$