3x2+10x=-7

To solve the equation \( 3x^2 + 10x = -7 \), first, we need to rearrange it into standard form:

1. Move \(-7\) to the left side of the equation: \[ 3x^2 + 10x + 7 = 0 \]

Now, we can solve this quadratic equation using the quadratic formula, which is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 3 \), \( b = 10 \), and \( c = 7 \).

2. Calculate the discriminant (\( b^2 - 4ac \)): \[ b^2 - 4ac = 10^2 - 4 \cdot 3 \cdot 7 = 100 - 84 = 16 \]

3. Now plug the values into the quadratic formula: \[ x = \frac{-10 \pm \sqrt{16}}{2 \cdot 3} \] \[ x = \frac{-10 \pm 4}{6} \]

4. Solve for the two possible values of \( x \):

   • For the positive root: \[ x = \frac{-10 + 4}{6} = \frac{-6}{6} = -1 \]
   • For the negative root: \[ x = \frac{-10 - 4}{6} = \frac{-14}{6} = -\frac{7}{3} \]

Thus, the solutions to the equation \( 3x^2 + 10x + 7 = 0 \) are: \[ x = -1 \quad \text{and} \quad x = -\frac{7}{3} \]