In 11 years, the age of Russel will be half of the square of the age he was 13 years ago. Find his present age.

x + 11 = ( 1 / 2 ) ( x - 13 ) ^ 2

x + 11 = ( 1 / 2 ) ( x ^ 2 - 2 x *13 + 13 ^ 2 )

x + 11 = ( 1 / 2 ) ( x ^ 2 - 26 x + 169 )

1 / 2 ( x ^ 2 ) - 13 x - x + 169 / 2 - 11 = 0

( 1 / 2 ) x ^ 2 - 13 x - x + 169 / 2 - 22 / 2 = 0

( 1 / 2 ) x ^ 2 - 13 x - x + 147 / 2 = 0

The exact solutions are:

7

or

21

In 11 years, the age of Russel will be half of the square of the age he was 13 YEARS AGO.

For that reason present age = 21

To solve this problem, let's break it down step by step.

Step 1: Assign variables
Let's assign a variable to Russel's present age. Let's call it "R."

Step 2: Set up the equation
According to the given information, in 11 years, Russel's age will be half of the square of his age 13 years ago. So, we can write the equation as:
(R + 11) = (1/2) * (R - 13)^2

Step 3: Solve the equation
To find Russel's present age, we need to solve this equation.

Expanding the equation, we have:
R + 11 = (1/2) * (R - 13) * (R - 13)

Simplifying further:
R + 11 = (1/2) * (R^2 - 26R + 169)
2R + 22 = R^2 - 26R + 169
0 = R^2 - 28R + 147

Step 4: Solve the quadratic equation
To solve the quadratic equation, we can use factoring or the quadratic formula. In this case, factoring is possible.

The equation factors as:
0 = (R - 7)(R - 21)

Setting each factor equal to zero, we find:
R - 7 = 0 → R = 7
R - 21 = 0 → R = 21

Step 5: Check the ages
Now, we will check which solution satisfies the given condition. Since the problem states that Russel's age will be half of the square of his age 13 years ago, we need to subtract 13 from his age.

For R = 7:
(R + 11) = (7 + 11) = 18
(R - 13)^2 = (7 - 13)^2 = (-6)^2 = 36
18 ≠ 36, so R = 7 is not a valid solution.

For R = 21:
(R + 11) = (21 + 11) = 32
(R - 13)^2 = (21 - 13)^2 = (8)^2 = 64

32 = 1/2 * 64, so R = 21 is a valid solution.

Therefore, Russel's present age is 21.