\begin{tabular}{l} What are the roots of the equation? \ $$ \qquad-3 x+19=-x^{2}+1 $$ \ Answer \ $$ \begin{array}{l}\frac{3+9 i}{2} \ \frac{-3+1 \sqrt{63}}{2} \ \frac{3+i \sqrt{63}}{2}\end{array} $$ \ \hline Answer \end{tabular}

Question

UpStudy AI · Accepted Answer

We start with the equation
$$
-3x+19=-x^2+1.
$$

**Step 1. Rearrange the equation**

Add $$x^2$$ to both sides and subtract $$1$$ from both sides:
$$
x^2-3x+19-1=0,
$$
which simplifies to
$$
x^2-3x+18=0.
$$

**Step 2. Compute the discriminant**

For a quadratic equation $$ax^2+bx+c=0$$, the discriminant is
$$
\Delta=b^2-4ac.
$$
Here, $$a=1$$, $$b=-3$$, and $$c=18$$. Thus,
$$
\Delta=(-3)^2-4\cdot1\cdot18=9-72=-63.
$$

**Step 3. Find the roots using the quadratic formula**

The quadratic formula is
$$
x=\frac{-b\pm\sqrt{\Delta}}{2a}.
$$
Substitute the values:
$$
x=\frac{-(-3)\pm\sqrt{-63}}{2\cdot1}=\frac{3\pm\sqrt{-63}}{2}.
$$
Since $$\sqrt{-63}=i\sqrt{63}$$, we have:
$$
x=\frac{3\pm i\sqrt{63}}{2}.
$$
This can also be written as:
$$
x=\frac{3\pm 3i\sqrt{7}}{2}.
$$

**Final Answer**

$$
x=\frac{3 \pm i\sqrt{63}}{2}\quad\text{or}\quad x=\frac{3 \pm 3i\sqrt{7}}{2}.
$$
The roots of the equation are $$ x = \frac{3 + i\sqrt{63}}{2} $$ and $$ x = \frac{3 - i\sqrt{63}}{2} $$.

\begin{tabular}{l} What are the roots of the equation? \\ \( \qquad-3 x+19=-x^{2}+1 \) \\ Answer \\ \( \begin{array}{l}\frac{3+9 i}{2} \\ \frac{-3+1 \sqrt{63}}{2} \\ \frac{3+i \sqrt{63}}{2}\end{array} \) \\ \hline Answer \end{tabular}

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