Solve for the roots in simplest form by completing the square: $$ -5 x^{2}-10 x-50=0 $$

Question

UpStudy AI · Accepted Answer

To solve the quadratic equation by completing the square:

$$
-5x^2 - 10x - 50 = 0
$$

**Step 1: Simplify the Equation**

First, divide the entire equation by $$-5$$ to make the coefficient of $$x^2$$ positive:

$$
x^2 + 2x + 10 = 0
$$

**Step 2: Move the Constant Term**

Subtract $$10$$ from both sides to isolate the quadratic and linear terms:

$$
x^2 + 2x = -10
$$

**Step 3: Complete the Square**

To complete the square, take half of the coefficient of $$x$$, which is $$1$$, and square it to get $$1$$. Add this to both sides:

$$
x^2 + 2x + 1 = -10 + 1
$$

$$
(x + 1)^2 = -9
$$

**Step 4: Solve for $$x$$**

Take the square root of both sides, remembering to include both the positive and negative roots:

$$
x + 1 = \pm \sqrt{-9}
$$

$$
x + 1 = \pm 3i
$$

Subtract $$1$$ from both sides to solve for $$x$$:

$$
x = -1 \pm 3i
$$

**Final Answer:**

The solutions are $$x = -1 + 3i$$ and $$x = -1 - 3i$$.
The solutions are $$x = -1 + 3i$$ and $$x = -1 - 3i$$.

Solve for the roots in simplest form by completing the square: \[ -5 x^{2}-10 x-50=0 \]