Questions: Solve for x: (x^2+6x+9)/(x+3)=0. Separate multiple answers with a comma and write DNE if there is no real answer.

Solution Steps

To solve the equation \(\frac{x^{2}+6x+9}{x+3}=0\), we need to find the values of \(x\) that make the numerator zero, since a fraction is zero when its numerator is zero (and the denominator is not zero).

1. Set the numerator equal to zero: \(x^2 + 6x + 9 = 0\).
2. Solve the quadratic equation \(x^2 + 6x + 9 = 0\) using the quadratic formula or by factoring.
3. Check that the solutions do not make the denominator zero.

We start with the equation

\[ \frac{x^{2}+6x+9}{x+3}=0. \]

To find the values of \(x\) that satisfy this equation, we need to set the numerator equal to zero:

\[ x^{2} + 6x + 9 = 0. \]

Next, we factor the quadratic equation:

\[ x^{2} + 6x + 9 = (x + 3)^{2} = 0. \]

This gives us the solution:

\[ x + 3 = 0 \implies x = -3. \]

Now, we must ensure that this solution does not make the denominator zero. The denominator is:

\[ x + 3. \]

Substituting \(x = -3\) into the denominator:

\[ -3 + 3 = 0. \]

Since the denominator is zero at \(x = -3\), this solution is not valid.

Final Answer