Questions: Solve x^2 + 9x = -14 x = (Separate answers by a comma. Write answers a reduced fractions.)

Solution Steps

To solve the quadratic equation \(x^2 + 9x = -14\), we first need to bring it to the standard form \(ax^2 + bx + c = 0\). Then, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions.

1. Rewrite the equation in standard form: \(x^2 + 9x + 14 = 0\).
2. Identify the coefficients \(a\), \(b\), and \(c\) from the standard form.
3. Use the quadratic formula to solve for \(x\).

We start with the equation:

\[ x^2 + 9x = -14 \]

To bring it to standard form, we add \(14\) to both sides:

\[ x^2 + 9x + 14 = 0 \]

From the standard form \(ax^2 + bx + c = 0\), we identify the coefficients:

\[ a = 1, \quad b = 9, \quad c = 14 \]

We calculate the discriminant \(D\) using the formula:

\[ D = b^2 - 4ac \]

Substituting the values:

\[ D = 9^2 - 4 \cdot 1 \cdot 14 = 81 - 56 = 25 \]

Using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{D}}{2a} \]

We substitute \(b\), \(D\), and \(a\):

\[ x = \frac{-9 \pm \sqrt{25}}{2 \cdot 1} = \frac{-9 \pm 5}{2} \]

Calculating the two possible values for \(x\):

1. For the positive root:

\[ x_1 = \frac{-9 + 5}{2} = \frac{-4}{2} = -2 \]

2. For the negative root:

\[ x_2 = \frac{-9 - 5}{2} = \frac{-14}{2} = -7 \]

Final Answer