The numbers are 45, 54
<em><u>Solution:</u></em>
Given a 2 digit number:
a = the 10's digit of the 1st number
b = units digit of the 1st number
From given information,
The sum of both digits, of either of two two-digit numbers, in whatever order the digits are written, is 9
a + b = 9 ---- eqn 1
The square of either of the digits of either number, minus the product of both digits, plus the square of the other digit is the number 21
Square of "a" - product of "a" and "b" + square of "b" = 21
--- eqn 2
<em><u>Let us solve eqn 1 and eqn 2</u></em>
From eqn 1,
a = 9 - b --- eqn 3
Substitute eqn 3 in eqn 2
![(b - 9)^2 - (9 - b)(b) + b^2 = 21\\\\b^2 + 81 - 18b - 9b + b^2 + b^2 = 21\\\\3b^2 - 27b + 60 = 0](https://tex.z-dn.net/?f=%28b%20-%209%29%5E2%20-%20%289%20-%20b%29%28b%29%20%2B%20b%5E2%20%3D%2021%5C%5C%5C%5Cb%5E2%20%2B%2081%20-%2018b%20-%209b%20%2B%20b%5E2%20%2B%20b%5E2%20%3D%2021%5C%5C%5C%5C3b%5E2%20-%2027b%20%2B%2060%20%3D%200)
Let us solve the above equation using quadratic formula,
![\text {For a quadratic equation } a x^{2}+b x+c=0, \text { where } a \neq 0\\\\x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}](https://tex.z-dn.net/?f=%5Ctext%20%7BFor%20a%20quadratic%20equation%20%7D%20a%20x%5E%7B2%7D%2Bb%20x%2Bc%3D0%2C%20%5Ctext%20%7B%20where%20%7D%20a%20%5Cneq%200%5C%5C%5C%5Cx%3D%5Cfrac%7B-b%20%5Cpm%20%5Csqrt%7Bb%5E%7B2%7D-4%20a%20c%7D%7D%7B2%20a%7D)
Using the above formula,
![\text{ for } 3b^2 - 27b + 60 = 0 , \text{ we have } a = 3 ; b = -27 ; c = 60](https://tex.z-dn.net/?f=%5Ctext%7B%20for%20%7D%203b%5E2%20-%2027b%20%2B%2060%20%3D%200%20%2C%20%5Ctext%7B%20we%20have%20%7D%20a%20%3D%203%20%3B%20b%20%3D%20-27%20%3B%20c%20%3D%2060)
<em><u>Substituting the values of a, b, c in above formula,</u></em>
![\begin{aligned}&b=\frac{-(-27) \pm \sqrt{(-27)^{2}-4(3)(60)}}{2 \times 3}\\\\&b=\frac{27 \pm \sqrt{729-720}}{6}\\\\&b=\frac{27 \pm \sqrt{9}}{6}\\\\&b=\frac{27 \pm 3}{6}\\\\&b=\frac{27+3}{6} \text { or } b=\frac{27-3}{6}\\\\&b=5 \text { or } b=4\end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%26b%3D%5Cfrac%7B-%28-27%29%20%5Cpm%20%5Csqrt%7B%28-27%29%5E%7B2%7D-4%283%29%2860%29%7D%7D%7B2%20%5Ctimes%203%7D%5C%5C%5C%5C%26b%3D%5Cfrac%7B27%20%5Cpm%20%5Csqrt%7B729-720%7D%7D%7B6%7D%5C%5C%5C%5C%26b%3D%5Cfrac%7B27%20%5Cpm%20%5Csqrt%7B9%7D%7D%7B6%7D%5C%5C%5C%5C%26b%3D%5Cfrac%7B27%20%5Cpm%203%7D%7B6%7D%5C%5C%5C%5C%26b%3D%5Cfrac%7B27%2B3%7D%7B6%7D%20%5Ctext%20%7B%20or%20%7D%20b%3D%5Cfrac%7B27-3%7D%7B6%7D%5C%5C%5C%5C%26b%3D5%20%5Ctext%20%7B%20or%20%7D%20b%3D4%5Cend%7Baligned%7D)
<em><u>Thus the required numbers are:</u></em>
when b = 5,
a = 9 - b
a = 9 - 5
a = 4
Thus the number is 45
When b = 4,
a = 9 - b
a = 9 - 4
a = 5
Thus the number is 54