.-.

If a + b=17

(a+b)^2 = 17^2 = 289 =a^2 +2ab + b^2

ab=72

:. 289 - 2ab = a^2 + b^2 = 289 - 2(72)

a^2 + b^2 = 145

Since $ab=72$ , $a=\frac{72}{b}$ . Therefore, $a+b=\frac{72}{b}+b=17$ . This means $b^{2}-17b+72=0$ . By the quadratic equation, $b=\frac{17 \pm \sqrt{17^{2}-4(1)(72)}}{2}=\frac{17}{2} \pm \frac{1}{2}$ . So $b=8, 9$ .

When $b=8$ we find $a$ to be $9$ . When $b=9$ , we find $a$ to be $8$ . Therefore, as $8^{2}+9^{2}=9^{2}+8^{2}$ , no matter what values we choose for $a$ and $b$ we choose, the answer will be $8^{2}+9^{2}=145$ .