Find two numbers whose sum is 27 and product is 182.
Step-by-Step Solution
Key Concept: Use the relationship between the sum and product of two numbers to form a quadratic equation. If the numbers are $x$ and $y$, then $x+y=S$ and $xy=P$. Substituting $y=S-x$ into $xy=P$ gives a quadratic in $x$ whose roots are the required numbers.
1. Let the two numbers be $x$ and $y$.
2. According to the problem,
$$\begin{cases} x+y = 27 \\ xy = 182 \end{cases}$$
3. From the first equation, express $y$ in terms of $x$:
$$y = 27 - x$$
4. Substitute this expression for $y$ in the product equation:
$$x(27 - x) = 182$$
5. Expand and bring all terms to one side:
$$27x - x^{2} = 182 \Rightarrow -x^{2} + 27x - 182 = 0$$
6. Multiply by $-1$ to obtain the standard quadratic form:
$$x^{2} - 27x + 182 = 0$$
7. Compute the discriminant $D$:
$$D = (-27)^{2} - 4\times1\times182 = 729 - 728 = 1$$
8. Since $D>0$, the equation has two real roots. Find the roots using the quadratic formula:
$$x = \frac{27 \pm \sqrt{1}}{2} = \frac{27 \pm 1}{2}$$
9. Hence,
$$x_{1} = \frac{27 + 1}{2} = 14, \quad x_{2} = \frac{27 - 1}{2} = 13$$
10. The corresponding values of $y$ are obtained from $y = 27 - x$:
- If $x = 14$, then $y = 27 - 14 = 13$.
- If $x = 13$, then $y = 27 - 13 = 14$.
11. Therefore, the two numbers are $13$ and $14$.
Correct Answer: The two numbers are 13 and 14.