How many $$ x $$-intercepts does the parabola with the following equation have? $$ y=2 x^{2}+14 x+22 $$ Show your work here Hint To add infinity (0), type "infinity"

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To determine the number of $$ x $$-intercepts of the parabola given by the equation

$$
y = 2x^2 + 14x + 22,
$$

we need to find the values of $$ x $$ for which $$ y = 0 $$. This means we need to solve the equation:

$$
2x^2 + 14x + 22 = 0.
$$

To find the number of solutions to this quadratic equation, we can use the discriminant, which is given by the formula:

$$
D = b^2 - 4ac,
$$

where $$ a $$, $$ b $$, and $$ c $$ are the coefficients from the quadratic equation $$ ax^2 + bx + c = 0 $$.

In our case, we have:

- $$ a = 2 $$
- $$ b = 14 $$
- $$ c = 22 $$

Now, we can calculate the discriminant:

$$
D = (14)^2 - 4(2)(22).
$$

Calculating $$ (14)^2 $$:

$$
14^2 = 196.
$$

Now calculating $$ 4(2)(22) $$:

$$
4 \cdot 2 \cdot 22 = 176.
$$

Now substituting these values into the discriminant formula:

$$
D = 196 - 176 = 20.
$$

Since the discriminant $$ D $$ is positive ($$ D > 0 $$), this indicates that the quadratic equation has two distinct real solutions. Therefore, the parabola intersects the $$ x $$-axis at two points.

Thus, the number of $$ x $$-intercepts of the parabola is

$$
\boxed{2}.
$$
The parabola has 2 $$ x $$-intercepts.

How many -intercepts does the parabola with the following equation have?

Show your work here
Hint To add infinity (0), type “infinity”

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How many -intercepts does the parabola with the following equation have? Show your work here Hint To add infinity (0), type “infinity”