-z^2-4z+7=0

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Solution for -z^2-4z+7=0 equation:


Simplifying
-1z2 + -4z + 7 = 0

Reorder the terms:
7 + -4z + -1z2 = 0

Solving
7 + -4z + -1z2 = 0

Solving for variable 'z'.

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-7 + 4z + z2 = 0

Move the constant term to the right:

Add '7' to each side of the equation.
-7 + 4z + 7 + z2 = 0 + 7

Reorder the terms:
-7 + 7 + 4z + z2 = 0 + 7

Combine like terms: -7 + 7 = 0
0 + 4z + z2 = 0 + 7
4z + z2 = 0 + 7

Combine like terms: 0 + 7 = 7
4z + z2 = 7

The z term is 4z.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4z + 4 + z2 = 7 + 4

Reorder the terms:
4 + 4z + z2 = 7 + 4

Combine like terms: 7 + 4 = 11
4 + 4z + z2 = 11

Factor a perfect square on the left side:
(z + 2)(z + 2) = 11

Calculate the square root of the right side: 3.31662479

Break this problem into two subproblems by setting 
(z + 2) equal to 3.31662479 and -3.31662479.

Subproblem 1

z + 2 = 3.31662479 Simplifying z + 2 = 3.31662479 Reorder the terms: 2 + z = 3.31662479 Solving 2 + z = 3.31662479 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = 3.31662479 + -2 Combine like terms: 2 + -2 = 0 0 + z = 3.31662479 + -2 z = 3.31662479 + -2 Combine like terms: 3.31662479 + -2 = 1.31662479 z = 1.31662479 Simplifying z = 1.31662479

Subproblem 2

z + 2 = -3.31662479 Simplifying z + 2 = -3.31662479 Reorder the terms: 2 + z = -3.31662479 Solving 2 + z = -3.31662479 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + z = -3.31662479 + -2 Combine like terms: 2 + -2 = 0 0 + z = -3.31662479 + -2 z = -3.31662479 + -2 Combine like terms: -3.31662479 + -2 = -5.31662479 z = -5.31662479 Simplifying z = -5.31662479

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.31662479, -5.31662479}

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