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Simplifying 5z2 + -12z = z2 + 144 Reorder the terms: -12z + 5z2 = z2 + 144 Reorder the terms: -12z + 5z2 = 144 + z2 Solving -12z + 5z2 = 144 + z2 Solving for variable 'z'. Reorder the terms: -144 + -12z + 5z2 + -1z2 = 144 + z2 + -144 + -1z2 Combine like terms: 5z2 + -1z2 = 4z2 -144 + -12z + 4z2 = 144 + z2 + -144 + -1z2 Reorder the terms: -144 + -12z + 4z2 = 144 + -144 + z2 + -1z2 Combine like terms: 144 + -144 = 0 -144 + -12z + 4z2 = 0 + z2 + -1z2 -144 + -12z + 4z2 = z2 + -1z2 Combine like terms: z2 + -1z2 = 0 -144 + -12z + 4z2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-36 + -3z + z2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-36 + -3z + z2)' equal to zero and attempt to solve: Simplifying -36 + -3z + z2 = 0 Solving -36 + -3z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '36' to each side of the equation. -36 + -3z + 36 + z2 = 0 + 36 Reorder the terms: -36 + 36 + -3z + z2 = 0 + 36 Combine like terms: -36 + 36 = 0 0 + -3z + z2 = 0 + 36 -3z + z2 = 0 + 36 Combine like terms: 0 + 36 = 36 -3z + z2 = 36 The z term is -3z. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3z + 2.25 + z2 = 36 + 2.25 Reorder the terms: 2.25 + -3z + z2 = 36 + 2.25 Combine like terms: 36 + 2.25 = 38.25 2.25 + -3z + z2 = 38.25 Factor a perfect square on the left side: (z + -1.5)(z + -1.5) = 38.25 Calculate the square root of the right side: 6.184658438 Break this problem into two subproblems by setting (z + -1.5) equal to 6.184658438 and -6.184658438.Subproblem 1
z + -1.5 = 6.184658438 Simplifying z + -1.5 = 6.184658438 Reorder the terms: -1.5 + z = 6.184658438 Solving -1.5 + z = 6.184658438 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = 6.184658438 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = 6.184658438 + 1.5 z = 6.184658438 + 1.5 Combine like terms: 6.184658438 + 1.5 = 7.684658438 z = 7.684658438 Simplifying z = 7.684658438Subproblem 2
z + -1.5 = -6.184658438 Simplifying z + -1.5 = -6.184658438 Reorder the terms: -1.5 + z = -6.184658438 Solving -1.5 + z = -6.184658438 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + z = -6.184658438 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + z = -6.184658438 + 1.5 z = -6.184658438 + 1.5 Combine like terms: -6.184658438 + 1.5 = -4.684658438 z = -4.684658438 Simplifying z = -4.684658438Solution
The solution to the problem is based on the solutions from the subproblems. z = {7.684658438, -4.684658438}Solution
z = {7.684658438, -4.684658438}
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