Question

listen what is the solution of the equation $3x^2 + 4x + 7 = 0$? if it has complex solutions, write them in the form $(a pm bi)$. use the keypad to enter the answer in the box. additional symbols can be found using the drop - down arrows at the top of the keypad. for this equation, the solutions of the equation are $x = \square$

Explanation:

Step1: Identify quadratic coefficients

For $3x^2 + 4x + 7 = 0$, $a=3$, $b=4$, $c=7$.

Step2: Calculate discriminant

$\Delta = b^2 - 4ac = 4^2 - 4(3)(7) = 16 - 84 = -68$

Step3: Apply quadratic formula

$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-4 \pm \sqrt{-68}}{6}$

Step4: Simplify complex root

$\sqrt{-68} = \sqrt{4 \times 17 \times (-1)} = 2i\sqrt{17}$, so $x = \frac{-4 \pm 2i\sqrt{17}}{6}$

Step5: Simplify the fraction

Factor out 2 in numerator: $x = \frac{2(-2 \pm i\sqrt{17})}{6} = \frac{-2 \pm i\sqrt{17}}{3}$
Rewrite as separate terms: $x = -\frac{2}{3} \pm \frac{\sqrt{17}}{3}i$

Answer:

$x = -\frac{2}{3} \pm \frac{\sqrt{17}}{3}i$