Find all solutions of the equation x2+2⁢x+19=0 by using the quadratic formula.

To solve the quadratic equation \( x^2 + 2x + 19 = 0 \) using the quadratic formula, we follow these steps:

### Step 1: Identify the coefficients
The quadratic formula is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

From the equation \( x^2 + 2x + 19 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = 2 \)
- \( c = 19 \)

### Step 2: Compute the discriminant
The discriminant is the part under the square root in the quadratic formula, given by \( b^2 - 4ac \).

\[
\text{Discriminant} = b^2 - 4ac = (2)^2 - 4(1)(19) = 4 - 76 = -72
\]

### Step 3: Apply the quadratic formula
Since the discriminant is negative, the solutions will involve complex numbers. Now substitute the values of \( a \), \( b \), and the discriminant into the quadratic formula:

\[
x = \frac{-2 \pm \sqrt{-72}}{2(1)} = \frac{-2 \pm \sqrt{72}i}{2}
\]

### Step 4: Simplify the expression
The square root of 72 can be simplified as:

\[
\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}
\]

Now substitute this back:

\[
x = \frac{-2 \pm 6\sqrt{2}i}{2}
\]

Finally, divide both terms by 2:

\[
x = -1 \pm 3\sqrt{2}i
\]

### Final Solution:
The solutions to the equation \( x^2 + 2x + 19 = 0 \) are:

\[
x = -1 + 3\sqrt{2}i \quad \text{and} \quad x = -1 - 3\sqrt{2}i
\]

Would you like further details on any of the steps, or have any questions?

### Related Questions:
1. What happens if the discriminant is positive in a quadratic equation?
2. How do complex solutions appear geometrically in the complex plane?
3. Can the quadratic formula be used for any degree polynomial?
4. How does completing the square compare with using the quadratic formula?
5. What are the applications of quadratic equations in physics or engineering?

### Tip:
Always check the discriminant first when using the quadratic formula to determine if the solutions will be real or complex!

Find all solutions of the equation x^2 + 2x + 19 = 0 by using the quadratic formula.

Explore how to solve the quadratic equation x^2 + 2x + 19 = 0 using the quadratic formula. This detailed guide explains each step, including identifying coefficients, calculating the discriminant, and finding complex solutions. Ideal for high school students learning algebra and complex numbers.

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