A woman is x years old. Her brother is four years older than her. The product of their ages is 1020. What is her age?

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The woman is x years old. As her brother is 4 years older than her, his age is x + 4.

The product of their ages is 1020.

So we have x*(x + 4) = 1020

x^2 + 4x = 1020

=> x^2 + 4x - 1020 = 0

=> x^2 + 34x - 30x - 1020 = 0

=> x( x + 34) - 30(x - 34) = 0

=> (x - 30)(x + 34) = 0

x = -34 and x = 30

We can eliminate the negative root.

**This gives the age of the woman as 30 years.**

Given that the woman's age is x.

Let us assume that the brother's age is y.

Given that the brothers age is 4 years older than her.

Then we will rewrite:

y= x+ 4 ................(1)

Also given that the product of their ages is 1020.

Then we will write:

x*y = 1020 ............(2)

Now we will use the substitution method to solve for x and y.

==> x*(x+4) = 1020

==> x^2 + 4x = 1020

==> x^2 + 4x - 1020 = 0

Now we will factor:

==> (x - 30 ) (x+34) = 0

==> x1= 30 ==> y1= 30+4 = 34

==> X2= -34 ( NOT VALID)

**Then the woman's age is 30 years old.**

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