For this problem, we are given a word problem, and we need to find the age of a character.
Grandma Mal's daughters had as many daughters as sisters.
This means that if Granda Mal's had "x" daughters, then the number of granddaughters must be:
[tex]y=(x-1)*x=x^2-x[/tex]Grandma Mal's age is the same as if you counted all of her daughters and granddaughters. This means that the sum between x and y is equal to her age:
[tex]x+y=\text{ age}[/tex]Grandma has already lived a half-century, but not three-quarters of one. This means that if we replace the first expression with the second one, we should have two inequalities. One that says that her age is greater or equal to 50 and one that says it's less than 75.
[tex]\begin{gathered} x^2+x\ge50\\ \\ x^2+x<75 \\ \end{gathered}[/tex][tex]\begin{gathered} x^2+x-50\ge0\\ \\ x_=\frac{-1\pm\sqrt{1^2-4(1)(-50)}}{2}=6.59\\ \\ x^2+x-75<0\\ \\ x=\frac{-1\pm1^2-4(1)(-75)}{2}=8.17\\ \end{gathered}[/tex]There are two possible values for x. It needs to be greater than 6.59 and less than 8.17. This is the number of daughters, therefore we can use fractional values, so we will use 6 and 8. Now we can find the number of granddaughters:
[tex]\begin{gathered} y=6*(6-1)=30\\ \\ y=8*(8-1)=56 \end{gathered}[/tex]The only ordered pair that fits the criteria is pair 56 and 8. Therefore she lived:
[tex]age=56+8=64[/tex]She's 64 years old.