Question: Fill in the blanks:
Play the game!
Find two numbers that multiply to make the 1st number and add to make the 2nd number.
Multiply 36; Add -12
______, and ______
1. The problem asks us to find two numbers that multiply to $$36$$ and add to $$-12$$.
2. Let the two numbers be $$x$$ and $$y$$.
3. We have the system:
$$x \times y = 36$$
$$x + y = -12$$
4. To find $$x$$ and $$y$$, we can use the quadratic equation approach. Consider the quadratic:
$$t^2 - (x+y)t + xy = 0$$
Substituting the sums and products:
$$t^2 - (-12)t + 36 = 0$$
$$t^2 + 12t + 36 = 0$$
5. Solve the quadratic equation:
$$t = \frac{-12 \pm \sqrt{12^2 - 4 \times 1 \times 36}}{2} = \frac{-12 \pm \sqrt{144 - 144}}{2} = \frac{-12 \pm 0}{2}$$
6. So, $$t = \frac{-12}{2} = -6$$.
7. Since the discriminant is zero, both numbers are equal:
$$x = y = -6$$.
8. Check:
$$-6 \times -6 = 36$$
$$-6 + (-6) = -12$$
Final answer: $$-6$$ and $$-6$$.