Solve the following equation by completing $$ x^{2}+6 x=-8 $$

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UpStudy AI · Accepted Answer

We start with the equation
$$
x^{2} + 6x = -8.
$$

**Step 1.** To complete the square, first add the square of half the coefficient of $$x$$ to both sides. The coefficient of $$x$$ is 6, so half of 6 is 3, and its square is
$$
3^2 = 9.
$$

**Step 2.** Add 9 to both sides:
$$
x^{2} + 6x + 9 = -8 + 9.
$$
This simplifies to:
$$
x^{2} + 6x + 9 = 1.
$$

**Step 3.** The left-hand side is now a perfect square:
$$
(x+3)^2 = 1.
$$

**Step 4.** Solve for $$x$$ by taking the square root of both sides:
$$
x+3 = \pm 1.
$$

**Step 5.** Solve for $$x$$ in each case:
- If $$x+3 = 1$$, then $$x = 1 - 3 = -2$$.
- If $$x+3 = -1$$, then $$x = -1 - 3 = -4$$.

Thus, the solutions to the equation are:
$$
x = -2 \quad \text{or} \quad x = -4.
$$
The solutions are $$ x = -2 $$ and $$ x = -4 $$.

Solve the following equation by completing

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