4z^2-7=-6z

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Solution for 4z^2-7=-6z equation:


Simplifying
4z2 + -7 = -6z

Reorder the terms:
-7 + 4z2 = -6z

Solving
-7 + 4z2 = -6z

Solving for variable 'z'.

Reorder the terms:
-7 + 6z + 4z2 = -6z + 6z

Combine like terms: -6z + 6z = 0
-7 + 6z + 4z2 = 0

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

Divide each side by '4'.
-1.75 + 1.5z + z2 = 0

Move the constant term to the right:

Add '1.75' to each side of the equation.
-1.75 + 1.5z + 1.75 + z2 = 0 + 1.75

Reorder the terms:
-1.75 + 1.75 + 1.5z + z2 = 0 + 1.75

Combine like terms: -1.75 + 1.75 = 0.00
0.00 + 1.5z + z2 = 0 + 1.75
1.5z + z2 = 0 + 1.75

Combine like terms: 0 + 1.75 = 1.75
1.5z + z2 = 1.75

The z term is 1.5z.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5z + 0.5625 + z2 = 1.75 + 0.5625

Reorder the terms:
0.5625 + 1.5z + z2 = 1.75 + 0.5625

Combine like terms: 1.75 + 0.5625 = 2.3125
0.5625 + 1.5z + z2 = 2.3125

Factor a perfect square on the left side:
(z + 0.75)(z + 0.75) = 2.3125

Calculate the square root of the right side: 1.520690633

Break this problem into two subproblems by setting 
(z + 0.75) equal to 1.520690633 and -1.520690633.

Subproblem 1

z + 0.75 = 1.520690633 Simplifying z + 0.75 = 1.520690633 Reorder the terms: 0.75 + z = 1.520690633 Solving 0.75 + z = 1.520690633 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + z = 1.520690633 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = 1.520690633 + -0.75 z = 1.520690633 + -0.75 Combine like terms: 1.520690633 + -0.75 = 0.770690633 z = 0.770690633 Simplifying z = 0.770690633

Subproblem 2

z + 0.75 = -1.520690633 Simplifying z + 0.75 = -1.520690633 Reorder the terms: 0.75 + z = -1.520690633 Solving 0.75 + z = -1.520690633 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + z = -1.520690633 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = -1.520690633 + -0.75 z = -1.520690633 + -0.75 Combine like terms: -1.520690633 + -0.75 = -2.270690633 z = -2.270690633 Simplifying z = -2.270690633

Solution

The solution to the problem is based on the solutions from the subproblems. z = {0.770690633, -2.270690633}

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