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Simplifying (2z + 6)(4z + -3) = (5z + 12) Reorder the terms: (6 + 2z)(4z + -3) = (5z + 12) Reorder the terms: (6 + 2z)(-3 + 4z) = (5z + 12) Multiply (6 + 2z) * (-3 + 4z) (6(-3 + 4z) + 2z * (-3 + 4z)) = (5z + 12) ((-3 * 6 + 4z * 6) + 2z * (-3 + 4z)) = (5z + 12) ((-18 + 24z) + 2z * (-3 + 4z)) = (5z + 12) (-18 + 24z + (-3 * 2z + 4z * 2z)) = (5z + 12) (-18 + 24z + (-6z + 8z2)) = (5z + 12) Combine like terms: 24z + -6z = 18z (-18 + 18z + 8z2) = (5z + 12) Reorder the terms: -18 + 18z + 8z2 = (12 + 5z) Remove parenthesis around (12 + 5z) -18 + 18z + 8z2 = 12 + 5z Solving -18 + 18z + 8z2 = 12 + 5z Solving for variable 'z'. Reorder the terms: -18 + -12 + 18z + -5z + 8z2 = 12 + 5z + -12 + -5z Combine like terms: -18 + -12 = -30 -30 + 18z + -5z + 8z2 = 12 + 5z + -12 + -5z Combine like terms: 18z + -5z = 13z -30 + 13z + 8z2 = 12 + 5z + -12 + -5z Reorder the terms: -30 + 13z + 8z2 = 12 + -12 + 5z + -5z Combine like terms: 12 + -12 = 0 -30 + 13z + 8z2 = 0 + 5z + -5z -30 + 13z + 8z2 = 5z + -5z Combine like terms: 5z + -5z = 0 -30 + 13z + 8z2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -3.75 + 1.625z + z2 = 0 Move the constant term to the right: Add '3.75' to each side of the equation. -3.75 + 1.625z + 3.75 + z2 = 0 + 3.75 Reorder the terms: -3.75 + 3.75 + 1.625z + z2 = 0 + 3.75 Combine like terms: -3.75 + 3.75 = 0.00 0.00 + 1.625z + z2 = 0 + 3.75 1.625z + z2 = 0 + 3.75 Combine like terms: 0 + 3.75 = 3.75 1.625z + z2 = 3.75 The z term is 1.625z. Take half its coefficient (0.8125). Square it (0.66015625) and add it to both sides. Add '0.66015625' to each side of the equation. 1.625z + 0.66015625 + z2 = 3.75 + 0.66015625 Reorder the terms: 0.66015625 + 1.625z + z2 = 3.75 + 0.66015625 Combine like terms: 3.75 + 0.66015625 = 4.41015625 0.66015625 + 1.625z + z2 = 4.41015625 Factor a perfect square on the left side: (z + 0.8125)(z + 0.8125) = 4.41015625 Calculate the square root of the right side: 2.100037202 Break this problem into two subproblems by setting (z + 0.8125) equal to 2.100037202 and -2.100037202.Subproblem 1
z + 0.8125 = 2.100037202 Simplifying z + 0.8125 = 2.100037202 Reorder the terms: 0.8125 + z = 2.100037202 Solving 0.8125 + z = 2.100037202 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.8125' to each side of the equation. 0.8125 + -0.8125 + z = 2.100037202 + -0.8125 Combine like terms: 0.8125 + -0.8125 = 0.0000 0.0000 + z = 2.100037202 + -0.8125 z = 2.100037202 + -0.8125 Combine like terms: 2.100037202 + -0.8125 = 1.287537202 z = 1.287537202 Simplifying z = 1.287537202Subproblem 2
z + 0.8125 = -2.100037202 Simplifying z + 0.8125 = -2.100037202 Reorder the terms: 0.8125 + z = -2.100037202 Solving 0.8125 + z = -2.100037202 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.8125' to each side of the equation. 0.8125 + -0.8125 + z = -2.100037202 + -0.8125 Combine like terms: 0.8125 + -0.8125 = 0.0000 0.0000 + z = -2.100037202 + -0.8125 z = -2.100037202 + -0.8125 Combine like terms: -2.100037202 + -0.8125 = -2.912537202 z = -2.912537202 Simplifying z = -2.912537202Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.287537202, -2.912537202}
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