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**Hint:**The accurate solution follows the concept of quadratic equations. By the use of quadratic equations, the equation can be solved very easily. The use of quadratic equations will help everyone to get through this simple question. Now let’s just solve the question through quadratic equations.

**Complete step-by-step solution:**

According to the question, the age of both the sisters is \[11\] years and \[14\] years respectively.

So, now assume after x years the product of their ages be taken as $(11 + x)\left( {14 + x} \right)$:

Therefore, we start with the equation:

\[ \Rightarrow (11 + x)\left( {14 + x} \right) = 304\]

\[ \Rightarrow 154 + 11x + 14x + {x^2} = 304\]

\[ \Rightarrow {x^2} + 25x - 150 = 0\]

\[ \Rightarrow \left( {x + 30} \right)(x - 5) = 0\]

\[ \Rightarrow x = - 30,5\]

Therefore, we can see that there are two results of \[x\]. But we have to calculate the number of years, and according to the result, there are two values. One is \[ - 30\], and as we know that years cannot come in negative values. So, this answer is not acceptable. The second one is \[5\], which is a positive number, and hence is acceptable.

**Therefore, in \[5\]years, the product of their ages will be \[304\].**

**Note:**Quadratic equations can be defined as the equation where the power of the variable is square so it can be written as \[{x^2}\]. It is very easy to solve a quadratic equation, and it is commonly written as (\[a{x^2} + bx + c = 0\]) , the quadratic equation in ‘x’ is an equation that can be written in a standard form, where ‘a’ ‘b’ ‘c’ are real numbers and \[a \ne 0\]. Here we could also use the quadratic formula and completing the square method to solve the quadratic equation.

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