Let's solve the problem step by step:
1. Define the variable:
Let [tex]\( x \)[/tex] be Gopi's present age.
2. Translate the problem statement into an equation:
According to the problem, the product of Gopi's age 5 years ago and his age 9 years later is 15.
Gopi's age 5 years ago: [tex]\( x - 5 \)[/tex]
Gopi's age 9 years later: [tex]\( x + 9 \)[/tex]
The product of these ages is given as:
[tex]\[
(x - 5)(x + 9) = 15
\][/tex]
3. Expand and simplify the equation:
[tex]\[
(x - 5)(x + 9) = x(x + 9) - 5(x + 9) = x^2 + 9x - 5x - 45 = x^2 + 4x - 45
\][/tex]
Therefore, the equation simplifies to:
[tex]\[
x^2 + 4x - 45 = 15
\][/tex]
4. Bring the equation to standard quadratic form:
[tex]\[
x^2 + 4x - 45 - 15 = 0
\][/tex]
[tex]\[
x^2 + 4x - 60 = 0
\][/tex]
5. Solve the quadratic equation:
We solve the quadratic equation [tex]\( x^2 + 4x - 60 = 0 \)[/tex] using the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 4 \)[/tex], and [tex]\( c = -60 \)[/tex].
Plug the values into the formula:
[tex]\[
x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-60)}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{-4 \pm \sqrt{16 + 240}}{2}
\][/tex]
[tex]\[
x = \frac{-4 \pm \sqrt{256}}{2}
\][/tex]
[tex]\[
x = \frac{-4 \pm 16}{2}
\][/tex]
6. Calculate the roots:
[tex]\[
x = \frac{-4 + 16}{2} = \frac{12}{2} = 6
\][/tex]
[tex]\[
x = \frac{-4 - 16}{2} = \frac{-20}{2} = -10
\][/tex]
7. Interpret the results:
We have two solutions: [tex]\( x = 6 \)[/tex] and [tex]\( x = -10 \)[/tex].
Since age cannot be negative, we discard [tex]\( x = -10 \)[/tex].
Therefore, Gopi's present age is [tex]\( 6 \)[/tex].
Answer: (a) 6
