Questions: Solve the equation using the quadratic formula. x^2 + 7x + 9 = 0 The solution set is

Transcript text: Solve the equation using the quadratic formula. \[ x^{2}+7 x+9=0 \] The solution set is $\square$

Solution

Solution Steps

To solve the quadratic equation $x^2 + 7x + 9 = 0$ using the quadratic formula, we identify the coefficients $a = 1$, $b = 7$, and $c = 9$. The quadratic formula is given by:

\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \]

We will calculate the discriminant $b^2 - 4ac$ and then use it to find the two possible values for $x$.

Step 1: Identify Coefficients

For the quadratic equation $x^2 + 7x + 9 = 0$, we identify the coefficients:

$a = 1$
$b = 7$
$c = 9$

Step 2: Calculate the Discriminant

We calculate the discriminant using the formula $D = b^2 - 4ac$: \[ D = 7^2 - 4 \cdot 1 \cdot 9 = 49 - 36 = 13 \]

Step 3: Apply the Quadratic Formula

Using the quadratic formula $x = \frac{{-b \pm \sqrt{D}}}{2a}$, we find the two solutions: \[ x_1 = \frac{{-7 + \sqrt{13}}}{2 \cdot 1} = \frac{{-7 + \sqrt{13}}}{2} \] \[ x_2 = \frac{{-7 - \sqrt{13}}}{2 \cdot 1} = \frac{{-7 - \sqrt{13}}}{2} \]

Step 4: Numerical Values

Calculating the numerical values: \[ x_1 \approx -1.6972 \quad \text{and} \quad x_2 \approx -5.3028 \]