Let's denote your sister's age as \( x \). Since your cousin is 26 years older than your sister, we can express her cousin's age as \( x + 26 \).
According to the problem, the product of their ages is 560. Therefore, we can write the equation:
\[ x(x + 26) = 560 \]
Expanding this gives:
\[ x^2 + 26x - 560 = 0 \]
Now, we can solve this quadratic equation using the quadratic formula, where \( a = 1 \), \( b = 26 \), and \( c = -560 \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Calculating the discriminant:
\[ b^2 - 4ac = 26^2 - 4(1)(-560) = 676 + 2240 = 2916 \]
Now, we take the square root of the discriminant:
\[ \sqrt{2916} = 54 \]
Now substituting back into the quadratic formula:
\[ x = \frac{-26 \pm 54}{2} \]
Calculating the two potential solutions for \( x \):
- \( x = \frac{28}{2} = 14 \)
- \( x = \frac{-80}{2} = -40 \) (not a valid solution since age cannot be negative)
So, your sister's age is:
\[ \boxed{14} \]
To check, if your sister is 14, then your cousin is \( 14 + 26 = 40 \). The product of their ages is \( 14 \times 40 = 560\), which confirms the solution is correct. Thus, your sister is indeed 14 years old.
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