Solving Quadratics With Complex Roots

Group all the terms on the same side of the equation by subtracting $10$ from both sides. Note that the discriminant can be evaluated before solving the equation. To do so, start by identifying the values of $a,$ $b,$ and $c.$ Next, substitute these values into the expression representing the discriminant — namely, $b^2-4ac.$ Since the discriminant is negative, the equation has no real solutions. In this context, it means that the jumper never reached $10$ meters above the lake after the rope became taut. In other words, the jumper was always more than $10$ meters above the lake. The solutions to the equation are $s_1=4-\sqrt{3}i$ and $s_2=4+\sqrt{3}i.$ Jordan's friends wondered how they could find the other solution without knowing the value of $K.$ Suddenly, Jordan remembered and shared with her friends the fact that she discovered last weekend. According to this information, the second solution is the conjugate of $x_1=5-2i.$ To find the conjugate of a complex number, change the sign of the imaginary part. Therefore, the second solution is $x_2=5+2i.$

The left-hand side expression is a sum of two squares and can be factored using complex numbers by using the following formula. In this case, $a=x$ and $b=3.$ Therefore, the given equation can be factored as follows. From here, the solutions to the given equation are $x_1=3i$ and $x_2=\text{-}3i.$ Consequently, Ali solved the equation correctly. In contrast, Jordan made a mistake when applying the formula. She did not rewrite $9$ as $3^2$ and used $9$ instead of $3$ for $b.$ Once again, the left-hand side expression is the sum of two squares and can be expressed as a product of two complex numbers. In this case, $a=x$ and $b=\sqrt{7}.$ Consequently, the given equation can be factored as follows. The solutions to the second equation are $x_1=\sqrt{7}i$ and $x_2=\text{-}\sqrt{7}i.$ Comparing the computations and solutions to the responses from the students, it can be seen that Jordan answered correctly this time. In contrast, Heichi made a mistake. Because he forgot to rewrite $7$ as $(\sqrt{7}{)}^2,$ he took $7$ instead of $\sqrt{7}$ for $b.$

The first equation that Jordan has to solve is $x^2-12ix-32=4.$ Since the coefficient of the $x\text{-}$term is negative, Jordan will use the second formula. To do so, she plans to group all the terms on the same side and rewrite each of them so that they resemble the formula. Comparing the left-hand side expression to the second formula, Jordan concluded that $b=6.$ Finally, she is ready to solve the first equation. From the resulting equation, Jordan concluded that the given equation has only one solution, $6i.$ Seeing how easily Jordan solved the first equation, Dominika began to worry. Next, Jordan noted that she needs to rewrite the constant term in terms of $\sqrt{10}.$ She realized that $10$ can be rewritten as $(\sqrt{10}{)}^2.$ Since the coefficient of the $x\text{-}$term is positive, Jordan will use the first formula written in the previous part. Jordan concluded that, as with the first equation, the second equation has only one solution, $x=\text{-}\sqrt{10}i.$ At this point, Dominika broke out in a cold sweat. After this, Jordan noticed that the linear term is $2(2i)x$ implying that $b=2i.$ Consequently, $b^2=\text{-}4.$ This led her to conclude that she needs to have $\text{-}4$ as a constant term somehow. Therefore, she decided to rewrite $\text{-}29$ as $\text{-}4-25.$ The first three terms can be rewritten as a perfect square trinomial with an imaginary term. Then, the resulting equation can be solved for $x.$ Finally, with a giant smile, Jordan concluded that the solutions to the third equation are $x_1=\text{-}5-2i$ and $x_2=5-2i.$ Therefore, Jordan won the challenge and Dominika is not happy at all!