4+z(2z+3)=15

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Solution for 4+z(2z+3)=15 equation:


Simplifying
4 + z(2z + 3) = 15

Reorder the terms:
4 + z(3 + 2z) = 15
4 + (3 * z + 2z * z) = 15
4 + (3z + 2z2) = 15

Solving
4 + 3z + 2z2 = 15

Solving for variable 'z'.

Reorder the terms:
4 + -15 + 3z + 2z2 = 15 + -15

Combine like terms: 4 + -15 = -11
-11 + 3z + 2z2 = 15 + -15

Combine like terms: 15 + -15 = 0
-11 + 3z + 2z2 = 0

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

Divide each side by '2'.
-5.5 + 1.5z + z2 = 0

Move the constant term to the right:

Add '5.5' to each side of the equation.
-5.5 + 1.5z + 5.5 + z2 = 0 + 5.5

Reorder the terms:
-5.5 + 5.5 + 1.5z + z2 = 0 + 5.5

Combine like terms: -5.5 + 5.5 = 0.0
0.0 + 1.5z + z2 = 0 + 5.5
1.5z + z2 = 0 + 5.5

Combine like terms: 0 + 5.5 = 5.5
1.5z + z2 = 5.5

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 = 5.5 + 0.5625

Reorder the terms:
0.5625 + 1.5z + z2 = 5.5 + 0.5625

Combine like terms: 5.5 + 0.5625 = 6.0625
0.5625 + 1.5z + z2 = 6.0625

Factor a perfect square on the left side:
(z + 0.75)(z + 0.75) = 6.0625

Calculate the square root of the right side: 2.46221445

Break this problem into two subproblems by setting 
(z + 0.75) equal to 2.46221445 and -2.46221445.

Subproblem 1

z + 0.75 = 2.46221445 Simplifying z + 0.75 = 2.46221445 Reorder the terms: 0.75 + z = 2.46221445 Solving 0.75 + z = 2.46221445 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 = 2.46221445 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = 2.46221445 + -0.75 z = 2.46221445 + -0.75 Combine like terms: 2.46221445 + -0.75 = 1.71221445 z = 1.71221445 Simplifying z = 1.71221445

Subproblem 2

z + 0.75 = -2.46221445 Simplifying z + 0.75 = -2.46221445 Reorder the terms: 0.75 + z = -2.46221445 Solving 0.75 + z = -2.46221445 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 = -2.46221445 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + z = -2.46221445 + -0.75 z = -2.46221445 + -0.75 Combine like terms: -2.46221445 + -0.75 = -3.21221445 z = -3.21221445 Simplifying z = -3.21221445

Solution

The solution to the problem is based on the solutions from the subproblems. z = {1.71221445, -3.21221445}

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