Slightly modified age problem

Let the current ages of Brenda and Jengis be $b$ and $j$ years old respectively. When Jengis was as old as Brenda now it was $j-b$ years ago. Then Brenda was $b-(j-b) = 2b-j$ years old. Therefore we have $b+j = 57$ $\implies \color{#3D99F6} j = 57-b$ ; and

$\begin{aligned} j & = (2b-j)^2 \\ \color{#3D99F6}57-b & = (2b-{\color{#3D99F6}(57-b)})^2 \\ 57-b & = (3b-57)^2 \\ 57-b & = 9x^2 -342x + 3249 \\ 9x^2 -341x + 3192 & = 0 \\ (9b-152)(b-21) & = 0 \\ \implies b & = 21 \\ j & = 57 - 21 = 36 \end{aligned}$

$b = \frac {152}9$ $\implies j = \frac {513}9$ are rejected because when Jengis was as old as Brenda now, Brenda was $2b - j < 0$ .

Now, when Jengis was 21, Brenda was $2b-j = 2(21)-36 = 6$ . The sum of their ages then $21+6 = \boxed{27}$ .

Let Brenda's current age be $y$ such that when Jengis was $y$ years old (Brenda's current age), Brenda was $x$ years old for some unknown $x$ to be determined. In other words, Jengis is always $(y-x)$ years older than Brenda. From what Brenda said, we also know that when Brenda is $y$ years at present, Jengis is $x^2$ years old.

From what Jengis said, we see that $x^2+y=57$ . Considering that Jengis is always $(y-x)$ years Brenda's senior as mentioned, we also see that $x^2=y+(y-x)=(1+1)y-x=2y-x$ . Rewriting this, we have:

$x^2+x=2y$

$y=\frac{x^2+x}{2}$

Considering also that $x^2+y=57$ :

$x^2+\frac{x^2+x}{2}=57$

$2x^2+x^2+x=2\times57$

$(2+1)x^2+x=114$

$3x^2+x=114$

$3x^2+x-114=0$

$3x^2-18x+19x-114=0$

$(x-6)(3x+19)=0$

$x=6$ or $x=-\frac{19}{3}$ (rejected as age is never negative)

$y=\frac{6^2+6}{2}=\frac{36+6}{2}=\frac{42}{2}=21$

This means Brenda is now 21 years old, a reasonable age for a young woman. When Jengis was 21 (Brenda's current age), Brenda was 6 years old, a little girl back then. This means their ages then summed to $21+6=\boxed{27}$ .

Since it is given that jengis age is a square number ,therefore we can narrow down his possible age to either 36 or 49 {The reason why I didn't consider 1,4,9,16or 25 is because it is given that jengis is older than brenda and if we took any of these ages , then that Condition would become invalid. It make more sense for jengis to be 36 than to be 49 ... And indeed 36 is the correct answer , if we consider 36 to be his age and apply condition number 2 to it , then we will find that indeed , Brenda current age is 21 .... 15 years back ..... Brenda =6 , jengis =21 ...Hence , The answer is 27

b + j = 57
j = [b - (j - b)]² = (2b - j)²
= [2(57 - j) - j]²
= (114 - 3j)²
= 9j² - 684j + 12996
0 = 9j² - 685j + 12996
= (j - 36)(9j - 361)
Since j > 57/2 = 28.5,
Then j = 36 and b = 21

Answer = 36 + 21 - 2(36 - 21)
= 57 - 2(15)
= 57 - 30
= 27