Sum Product 1D967F

1. **State the problem:** Find two numbers whose sum is $\frac{1}{4}$ and whose product is $-1.8$. 2. **Use the formulas:** Let the two numbers be $x$ and $y$. Then: $$x + y = \frac{1}{4}$$ $$xy = -1.8$$ 3. **Form a quadratic equation:** Using the sum and product of roots, the quadratic equation is: $$t^2 - (x+y)t + xy = 0$$ Substitute the values: $$t^2 - \frac{1}{4}t - 1.8 = 0$$ 4. **Solve the quadratic equation:** Use the quadratic formula: $$t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-\frac{1}{4}$, and $c=-1.8$. Calculate the discriminant: $$\Delta = \left(-\frac{1}{4}\right)^2 - 4(1)(-1.8) = \frac{1}{16} + 7.2 = \frac{1}{16} + \frac{115.2}{16} = \frac{116.2}{16}$$ Calculate the roots: $$t = \frac{\frac{1}{4} \pm \sqrt{\frac{116.2}{16}}}{2} = \frac{\frac{1}{4} \pm \frac{\sqrt{116.2}}{4}}{2} = \frac{1 \pm \sqrt{116.2}}{8}$$ 5. **Approximate the roots:** $$\sqrt{116.2} \approx 10.78$$ So, $$t_1 = \frac{1 + 10.78}{8} = \frac{11.78}{8} = 1.4725$$ $$t_2 = \frac{1 - 10.78}{8} = \frac{-9.78}{8} = -1.2225$$ 6. **Answer:** The two numbers are approximately $1.4725$ and $-1.2225$. These satisfy: $$1.4725 + (-1.2225) = 0.25 = \frac{1}{4}$$ $$1.4725 \times (-1.2225) \approx -1.8$$