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Answer:
> Therefore, the brother is approximately 21.085 years old
> The girl is approximately 3.085 years old.
Explanation:
• Since the girl is 18 years younger than her brother, her age will be ( x - 18 ).
The problem states that the product of their ages is 65. This can be written as:
[tex]x(x - 18) = 65[/tex]
Now, let's solve this quadratic equation:
[tex]x^2 - 18x = 65[/tex]
[tex]x^2 - 18x - 65 = 0[/tex]
We can solve this using the quadratic formula, which is:
[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
[tex]{( x^2 - 18x - 65 = 0 ), \text {the coefficients are:}}[/tex]
[tex] a = 1, \quad b = -18, \quad c = -65[/tex]
Plugging these into the quadratic formula:
[tex]x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4 \cdot 1 \cdot (-65)}}{2 \cdot 1}[/tex]
[tex]x = \frac{18 \pm \sqrt{324 + 260}}{2}[/tex]
[tex]x = \frac{18 \pm \sqrt{584}}{2}[/tex]
[tex]x = \frac{18 \pm 24.17}{2}[/tex]
This gives us two possible solutions:
[tex]x = \frac{18 + 24.17}{2} = \frac{42.17}{2} = \boxed {21.085}[/tex]
[tex] x = \frac{18 - 24.17}{2} = \frac{-6.17}{2} = \boxed {-3.085}[/tex]