To solve for Jon's age [tex]\( j \)[/tex], we start with the information provided:
- Jon is 3 years younger than Laura.
- The product of their ages is 1,330.
- If [tex]\( j \)[/tex] represents Jon's age, then [tex]\( j + 3 \)[/tex] represents Laura's age.
First, set up the equation representing the product of their ages:
[tex]\[ j \times (j + 3) = 1330 \][/tex]
Next, simplify and rearrange the equation to form a standard quadratic equation:
[tex]\[ j^2 + 3j - 1330 = 0 \][/tex]
At this step, we need to solve the quadratic equation for [tex]\( j \)[/tex].
Using the quadratic formula [tex]\( j = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\(a = 1\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(c = -1330\)[/tex]:
Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac = 3^2 - 4(1)(-1330) = 9 + 5320 = 5329 \][/tex]
Take the square root of the discriminant:
[tex]\[ \sqrt{5329} = 73 \][/tex]
Plug the values back into the quadratic formula:
[tex]\[ j = \frac{-3 \pm 73}{2} \][/tex]
This results in two potential solutions:
[tex]\[ j = \frac{-3 + 73}{2} = \frac{70}{2} = 35 \][/tex]
[tex]\[ j = \frac{-3 - 73}{2} = \frac{-76}{2} = -38 \][/tex]
Since ages cannot be negative, discard the negative solution. Therefore, Jon's age is:
[tex]\[ j = 35 \][/tex]
So the value of [tex]\( j \)[/tex] could be Jon's age is:
[tex]\[ \boxed{35} \][/tex]
