Solve the following equation using the quadratic formula. $$ -6 x^{2}=-3 x+14 $$

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UpStudy AI · Accepted Answer

To solve the equation $$ -6 x^{2} = -3 x + 14 $$ using the quadratic formula, we first need to rearrange it into the standard form $$ ax^2 + bx + c = 0 $$.

1. **Rearranging the equation:**
   $$
   -6x^2 + 3x - 14 = 0
   $$
   This gives us:
   - $$ a = -6 $$
   - $$ b = 3 $$
   - $$ c = -14 $$

2. **Using the quadratic formula:**
   The quadratic formula is given by:
   $$
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   $$

3. **Calculating the discriminant:**
   First, we calculate $$ b^2 - 4ac $$:
   $$
   b^2 - 4ac = 3^2 - 4(-6)(-14)
   $$

Now, let's calculate this step by step.

- Calculate $$ 3^2 $$:
  $$
  3^2 = 9
  $$

- Calculate $$ 4 \times -6 \times -14 $$:
  $$
  4 \times -6 \times -14 = 336
  $$

- Now, substitute these values into the discriminant:
  $$
  9 - 336 = -327
  $$

Since the discriminant is negative, this indicates that the equation has no real solutions, but two complex solutions.

4. **Calculating the solutions:**
   We will now calculate the complex solutions using the quadratic formula:
   $$
   x = \frac{-3 \pm \sqrt{-327}}{2 \times -6}
   $$

Let's calculate this.

- The square root of $$-327$$ can be expressed as:
  $$
  \sqrt{-327} = i\sqrt{327}
  $$

Now, we can substitute this back into the formula:
$$
x = \frac{-3 \pm i\sqrt{327}}{-12}
$$

This simplifies to:
$$
x = \frac{3}{12} \mp \frac{i\sqrt{327}}{12}
$$
$$
x = \frac{1}{4} \mp \frac{i\sqrt{327}}{12}
$$

Thus, the solutions to the equation $$ -6 x^{2} = -3 x + 14 $$ are:
$$
x = \frac{1}{4} + \frac{i\sqrt{327}}{12} \quad \text{and} \quad x = \frac{1}{4} - \frac{i\sqrt{327}}{12}
$$
The solutions are $$ x = \frac{1}{4} + \frac{i\sqrt{327}}{12} $$ and $$ x = \frac{1}{4} - \frac{i\sqrt{327}}{12} $$.

Solve the following equation using the quadratic formula. \( -6 x^{2}=-3 x+14 \)

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