Questions: Find all complex solutions of 7x^2 + 7x + 3 = 0. (If there is more than one solution, separate them with commas.)

Transcript text: Find all. complex solutions of $7 x^{2}+7 x+3=0$. (If there is more than one solution, separate them with commas.)

Solution

Solution Steps

To find the complex solutions of the quadratic equation $7x^2 + 7x + 3 = 0$, we can use the quadratic formula, which is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Here, $a = 7$, $b = 7$, and $c = 3$. We will calculate the discriminant $b^2 - 4ac$ to determine the nature of the roots and then apply the formula to find the solutions.

Step 1: Identify the Coefficients

For the quadratic equation $7x^2 + 7x + 3 = 0$, we identify the coefficients as follows:

$a = 7$
$b = 7$
$c = 3$

Step 2: Calculate the Discriminant

The discriminant $D$ is calculated using the formula: \[ D = b^2 - 4ac \] Substituting the values, we have: \[ D = 7^2 - 4 \cdot 7 \cdot 3 = 49 - 84 = -35 \]

Step 3: Determine the Solutions

Since the discriminant is negative ($D = -35$), the solutions will be complex. We use the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] Substituting the values, we find: \[ x = \frac{-7 \pm \sqrt{-35}}{2 \cdot 7} = \frac{-7 \pm i\sqrt{35}}{14} \] This simplifies to: \[ x = -\frac{1}{2} \pm \frac{\sqrt{35}}{14}i \]