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Simplifying -4z(z + -3) = 2 + -3z Reorder the terms: -4z(-3 + z) = 2 + -3z (-3 * -4z + z * -4z) = 2 + -3z (12z + -4z2) = 2 + -3z Solving 12z + -4z2 = 2 + -3z Solving for variable 'z'. Reorder the terms: -2 + 12z + 3z + -4z2 = 2 + -3z + -2 + 3z Combine like terms: 12z + 3z = 15z -2 + 15z + -4z2 = 2 + -3z + -2 + 3z Reorder the terms: -2 + 15z + -4z2 = 2 + -2 + -3z + 3z Combine like terms: 2 + -2 = 0 -2 + 15z + -4z2 = 0 + -3z + 3z -2 + 15z + -4z2 = -3z + 3z Combine like terms: -3z + 3z = 0 -2 + 15z + -4z2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 0.5 + -3.75z + z2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -3.75z + -0.5 + z2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -3.75z + z2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -3.75z + z2 = 0 + -0.5 -3.75z + z2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -3.75z + z2 = -0.5 The z term is -3.75z. Take half its coefficient (-1.875). Square it (3.515625) and add it to both sides. Add '3.515625' to each side of the equation. -3.75z + 3.515625 + z2 = -0.5 + 3.515625 Reorder the terms: 3.515625 + -3.75z + z2 = -0.5 + 3.515625 Combine like terms: -0.5 + 3.515625 = 3.015625 3.515625 + -3.75z + z2 = 3.015625 Factor a perfect square on the left side: (z + -1.875)(z + -1.875) = 3.015625 Calculate the square root of the right side: 1.736555499 Break this problem into two subproblems by setting (z + -1.875) equal to 1.736555499 and -1.736555499.Subproblem 1
z + -1.875 = 1.736555499 Simplifying z + -1.875 = 1.736555499 Reorder the terms: -1.875 + z = 1.736555499 Solving -1.875 + z = 1.736555499 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + z = 1.736555499 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + z = 1.736555499 + 1.875 z = 1.736555499 + 1.875 Combine like terms: 1.736555499 + 1.875 = 3.611555499 z = 3.611555499 Simplifying z = 3.611555499Subproblem 2
z + -1.875 = -1.736555499 Simplifying z + -1.875 = -1.736555499 Reorder the terms: -1.875 + z = -1.736555499 Solving -1.875 + z = -1.736555499 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '1.875' to each side of the equation. -1.875 + 1.875 + z = -1.736555499 + 1.875 Combine like terms: -1.875 + 1.875 = 0.000 0.000 + z = -1.736555499 + 1.875 z = -1.736555499 + 1.875 Combine like terms: -1.736555499 + 1.875 = 0.138444501 z = 0.138444501 Simplifying z = 0.138444501Solution
The solution to the problem is based on the solutions from the subproblems. z = {3.611555499, 0.138444501}
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