Age Calculation: Solving Meena's Age Problem

This problem asks us to find Meena's current age based on a relationship between her age and its square. We can solve this using algebra by setting up and solving an equation.

Step 1: Define the Variable

Let's use a variable to represent Meena's age. Let '$a$' be Meena's age in years.

Step 2: Formulate the Equation

The problem states: "The square of Meena's age is $9$ more than $8$ times her age." Let's translate this into a mathematical equation:

• The square of Meena's age is $a^2$.
• $8$ times her age is $8a$.
• $9$ more than $8$ times her age is $8a + 9$.

So, the equation is:

$a^2 = 8a + 9$

Step 3: Rearrange into a Quadratic Equation

To solve this equation, we need to rearrange it into the standard quadratic form, which is $ax^2 + bx + c = 0$. We do this by moving all terms to one side of the equation:

Subtract $8a$ and $9$ from both sides:

$a^2 - 8a - 9 = 0$

Step 4: Solve the Quadratic Equation

We can solve this quadratic equation by factoring. We need to find two numbers that multiply to $-9$ and add up to $-8$. These two numbers are $-9$ and $1$.

Therefore, we can factor the equation as follows:

$(a - 9)(a + 1) = 0$

For the product of two factors to be zero, at least one of the factors must be zero. This gives us two possible solutions:

1. $a - 9 = 0 \implies a = 9$
2. $a + 1 = 0 \implies a = -1$

Step 5: Interpret the Solution

We found two possible values for Meena's age: $9$ and $-1$. Since age cannot be negative, we discard the solution $a = -1$.

Thus, Meena's age is $9$ years.

Step 6: Verify the Answer

Let's check if Meena's age being $9$ satisfies the original condition:

• Square of her age: $9^2 = 81$.
• $8$ times her age: $8 \times 9 = 72$.
• $9$ more than $8$ times her age: $72 + 9 = 81$.

Since $81$ is equal to $81$, our solution is correct. Meena's age is $9$ years.