Your cousin is 26 years older than your sister the product of there ages is 560. How old is your sister?

Answer 1

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 $x$:

1. $x=\frac{28}{2}=14$
2. $x=\frac{-80}{2}=-40$ (not a valid solution since age cannot be negative)

So, your sister's age is:

$$\boxed{{\displaystyle 14}}$$

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.