Question Examples What are the roots of the equation? $$ -8 x+30=-3 x^{2}+5 x $$ Answer $$ \frac{-13 \pm 1 / \sqrt{191}}{6} $$ subuit 6 and - $$ \frac{8}{8} $$ $$ \frac{1821 \sqrt{151}}{6} $$ $$ \frac{5}{8} $$ and -6

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UpStudy AI · Accepted Answer

$$
-8x + 30 = -3x^2 + 5x
$$

1. **Isolate all terms on one side:**  
   Add $$8x$$ to both sides:
   $$
   30 = -3x^2 + 13x
   $$
   Subtract $$30$$ from both sides:
   $$
   0 = -3x^2 + 13x - 30
   $$

2. **Multiply by $$-1$$ to simplify:**
   $$
   0 = 3x^2 - 13x + 30
   $$

3. **Apply the quadratic formula:**  
   For an equation $$ax^2+bx+c=0$$, the roots are
   $$
   x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
   $$
   Here, $$a = 3$$, $$b = -13$$, and $$c = 30$$. Substitute these values:
   $$
   x = \frac{-(-13) \pm \sqrt{(-13)^2 - 4\cdot 3 \cdot 30}}{2\cdot 3}
   $$
   Simplify:
   $$
   x = \frac{13 \pm \sqrt{169 - 360}}{6}
   $$
   $$
   x = \frac{13 \pm \sqrt{-191}}{6}
   $$

4. **Express the square root of the negative number using $$i$$:**
   $$
   \sqrt{-191} = i\sqrt{191}
   $$
   So the roots become:
   $$
   x = \frac{13 \pm i\sqrt{191}}{6}
   $$

Thus, the roots of the equation are:
$$
\boxed{\frac{13 \pm i\sqrt{191}}{6}}
$$
The roots of the equation are $$ \frac{13 \pm i\sqrt{191}}{6} $$.

Question Examples What are the roots of the equation? \[ -8 x+30=-3 x^{2}+5 x \] Answer \( \frac{-13 \pm 1 / \sqrt{191}}{6} \) subuit 6 and - \( \frac{8}{8} \) \( \frac{1821 \sqrt{151}}{6} \) \( \frac{5}{8} \) and -6

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