Homework

- 16 x 9 = -144

- 16 + 9 = - 7

The product of two integers is -144.
The sum of the integers is -7.
What are the two integers?

$\begin{array}{|rcll|} \hline n_1 + n_2 &=& -7 \\ n_2 &=& -7 - n_1 \\\\ n_1\cdot n_2 &=& -144 \\ n_1\cdot (-7 - n_1) &=& -144 \\ -7\cdot n_1 - n_1^2 &=& -144 \quad & | \quad \cdot(-1)\\ 7\cdot n_1 + n_1^2 &=& 144 \quad & | \quad -144\\ 7\cdot n_1 + n_1^2 -144 &=& 0\\ n_1^2 + 7 n_1 -144 &=& 0\\ n_{1,2} &=& \frac{-7\pm\sqrt{(-7)^2-4\cdot(-144)} }{2} \\ n_{1,2} &=& \frac{-7\pm\sqrt{49+4\cdot 144} }{2} \\ n_{1,2} &=& \frac{-7\pm\sqrt{625} }{2} \quad & | \quad \sqrt{625} = 25\\ n_{1,2} &=& \frac{-7\pm 25 }{2}\\ n_1 = \frac{-7+25}{2} &\text{or}& n_1 = \frac{-7-25}{2} \\ n_1 = 9 &\text{or}& n_1 = -16 \\\\ n_2 = -7 - n_1 &\text{or}& n_2 = -7 - n_1 \\ n_2 = -7 - 9 &\text{or}& n_2 = -7 - (-16) \\ n_2 = -16 &\text{or}& n_2 = -7 +16 \\ & & n_2 =9 \\ \hline \end{array}$

The two integers are 9 and -16