z^2+z-25=0

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Solution for z^2+z-25=0 equation:


Simplifying
z2 + z + -25 = 0

Reorder the terms:
-25 + z + z2 = 0

Solving
-25 + z + z2 = 0

Solving for variable 'z'.

Begin completing the square.

Move the constant term to the right:

Add '25' to each side of the equation.
-25 + z + 25 + z2 = 0 + 25

Reorder the terms:
-25 + 25 + z + z2 = 0 + 25

Combine like terms: -25 + 25 = 0
0 + z + z2 = 0 + 25
z + z2 = 0 + 25

Combine like terms: 0 + 25 = 25
z + z2 = 25

The z term is z.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
z + 0.25 + z2 = 25 + 0.25

Reorder the terms:
0.25 + z + z2 = 25 + 0.25

Combine like terms: 25 + 0.25 = 25.25
0.25 + z + z2 = 25.25

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = 25.25

Calculate the square root of the right side: 5.024937811

Break this problem into two subproblems by setting 
(z + 0.5) equal to 5.024937811 and -5.024937811.

Subproblem 1

z + 0.5 = 5.024937811 Simplifying z + 0.5 = 5.024937811 Reorder the terms: 0.5 + z = 5.024937811 Solving 0.5 + z = 5.024937811 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = 5.024937811 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = 5.024937811 + -0.5 z = 5.024937811 + -0.5 Combine like terms: 5.024937811 + -0.5 = 4.524937811 z = 4.524937811 Simplifying z = 4.524937811

Subproblem 2

z + 0.5 = -5.024937811 Simplifying z + 0.5 = -5.024937811 Reorder the terms: 0.5 + z = -5.024937811 Solving 0.5 + z = -5.024937811 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + z = -5.024937811 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + z = -5.024937811 + -0.5 z = -5.024937811 + -0.5 Combine like terms: -5.024937811 + -0.5 = -5.524937811 z = -5.524937811 Simplifying z = -5.524937811

Solution

The solution to the problem is based on the solutions from the subproblems. z = {4.524937811, -5.524937811}

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