(x+1)^2=7918-2x

To solve the equation \((x+1)^2 = 7918 - 2x\), we will first expand the left-hand side and then rearrange the equation.

1. Expand \((x + 1)^2\): \[ (x + 1)^2 = x^2 + 2x + 1 \]

2. Substitute this back into the equation: \[ x^2 + 2x + 1 = 7918 - 2x \]

3. Bring all terms to one side of the equation: \[ x^2 + 2x + 1 + 2x - 7918 = 0 \] \[ x^2 + 4x + 1 - 7918 = 0 \] \[ x^2 + 4x - 7917 = 0 \]

4. Now, we will apply the quadratic formula where \(a = 1\), \(b = 4\), and \(c = -7917\): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

5. Calculate the discriminant (\(b^2 - 4ac\)): \[ b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot (-7917) \] \[ = 16 + 31668 \] \[ = 31684 \]

6. Now, calculate \(x\): \[ x = \frac{-4 \pm \sqrt{31684}}{2 \cdot 1} \] \[ = \frac{-4 \pm 178}{2} \]

7. Calculate the two possible values for \(x\):

   • First solution: \[ x = \frac{174}{2} = 87 \]
   • Second solution: \[ x = \frac{-182}{2} = -91 \]

Thus, the solutions to the equation \((x + 1)^2 = 7918 - 2x\) are: \[ x = 87 \quad \text{and} \quad x = -91 \]