3 The sum of two numbers is -2 . Their product is -24 . What are the two numbers?

Question

UpStudy AI · Accepted Answer

Let the two numbers be $$ x $$ and $$ y $$.

1. From the problem, we have the equations:
   $$
   x + y = -2
   $$
   $$
   xy = -24
   $$

2. We can express $$ y $$ in terms of $$ x $$ from the first equation:
   $$
   y = -2 - x
   $$

3. Substitute this expression into the product equation:
   $$
   x(-2 - x) = -24
   $$
   Simplify:
   $$
   -2x - x^2 = -24
   $$

4. Rearrange the equation to standard quadratic form by adding $$ 24 $$ to both sides and multiplying through by $$-1$$:
   $$
   x^2 + 2x - 24 = 0
   $$

5. Factor the quadratic:
   Look for two numbers whose product is $$-24$$ and whose sum is $$2$$. The numbers $$6$$ and $$-4$$ work since:
   $$
   6 \times (-4) = -24 \quad \text{and} \quad 6 + (-4) = 2
   $$
   So, the factored form is:
   $$
   (x + 6)(x - 4) = 0
   $$

6. Solve for $$ x $$:
   $$
   x + 6 = 0 \quad \Longrightarrow \quad x = -6
   $$
   $$
   x - 4 = 0 \quad \Longrightarrow \quad x = 4
   $$

7. Find the corresponding $$ y $$ for each $$ x $$:
   - If $$ x = -6 $$, then:
     $$
     y = -2 - (-6) = 4
     $$
     
   - If $$ x = 4 $$, then:
     $$
     y = -2 - 4 = -6
     $$

Thus, the two numbers are $$ 4 $$ and $$-6$$.
The two numbers are 4 and -6.

3 The sum of two numbers is -2 . Their product is -24 . What are the two numbers?

Upstudy AI Solution

Answer

Solution

Bonus Knowledge

Related Questions

Latest Algebra Questions