Sure, let's work through the problem step by step to find Jon's age.
1. Define Variables:
- Let [tex]\( j \)[/tex] represent Jon's age.
- Since Laura is 3 years older than Jon, Laura's age is [tex]\( j + 3 \)[/tex].
2. Set Up the Equation:
- According to the problem, the product of their ages is 1330. Therefore, we can write the equation:
[tex]\[
j \times (j + 3) = 1330
\][/tex]
3. Solve the Equation:
- We need to solve the quadratic equation:
[tex]\[
j^2 + 3j - 1330 = 0
\][/tex]
- To solve this quadratic equation, we can use 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].
- Plugging in the values we get:
[tex]\[
j = \frac{{-3 \pm \sqrt{{3^2 - 4 \cdot 1 \cdot (-1330)}}}}{2 \cdot 1}
\][/tex]
[tex]\[
j = \frac{{-3 \pm \sqrt{{9 + 5320}}}}{2}
\][/tex]
[tex]\[
j = \frac{{-3 \pm \sqrt{{5329}}}}{2}
\][/tex]
- The square root of 5329 is 73, so:
[tex]\[
j = \frac{{-3 \pm 73}}{2}
\][/tex]
- This leads to two possible solutions:
[tex]\[
j = \frac{{-3 + 73}}{2} = \frac{70}{2} = 35
\][/tex]
[tex]\[
j = \frac{{-3 - 73}}{2} = \frac{-76}{2} = -38
\][/tex]
4. Determine the Reasonable Solution:
- Jon's age must be a non-negative number. Therefore, the negative solution of [tex]\( -38 \)[/tex] is not reasonable in this context.
- Thus, the only feasible solution is:
[tex]\[
j = 35
\][/tex]
So, Jon's age is [tex]\( \boxed{35} \)[/tex].
