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Simplifying 4z(3z + 6) = 90 Reorder the terms: 4z(6 + 3z) = 90 (6 * 4z + 3z * 4z) = 90 (24z + 12z2) = 90 Solving 24z + 12z2 = 90 Solving for variable 'z'. Reorder the terms: -90 + 24z + 12z2 = 90 + -90 Combine like terms: 90 + -90 = 0 -90 + 24z + 12z2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-15 + 4z + 2z2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-15 + 4z + 2z2)' equal to zero and attempt to solve: Simplifying -15 + 4z + 2z2 = 0 Solving -15 + 4z + 2z2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -7.5 + 2z + z2 = 0 Move the constant term to the right: Add '7.5' to each side of the equation. -7.5 + 2z + 7.5 + z2 = 0 + 7.5 Reorder the terms: -7.5 + 7.5 + 2z + z2 = 0 + 7.5 Combine like terms: -7.5 + 7.5 = 0.0 0.0 + 2z + z2 = 0 + 7.5 2z + z2 = 0 + 7.5 Combine like terms: 0 + 7.5 = 7.5 2z + z2 = 7.5 The z term is 2z. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2z + 1 + z2 = 7.5 + 1 Reorder the terms: 1 + 2z + z2 = 7.5 + 1 Combine like terms: 7.5 + 1 = 8.5 1 + 2z + z2 = 8.5 Factor a perfect square on the left side: (z + 1)(z + 1) = 8.5 Calculate the square root of the right side: 2.915475947 Break this problem into two subproblems by setting (z + 1) equal to 2.915475947 and -2.915475947.Subproblem 1
z + 1 = 2.915475947 Simplifying z + 1 = 2.915475947 Reorder the terms: 1 + z = 2.915475947 Solving 1 + z = 2.915475947 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + z = 2.915475947 + -1 Combine like terms: 1 + -1 = 0 0 + z = 2.915475947 + -1 z = 2.915475947 + -1 Combine like terms: 2.915475947 + -1 = 1.915475947 z = 1.915475947 Simplifying z = 1.915475947Subproblem 2
z + 1 = -2.915475947 Simplifying z + 1 = -2.915475947 Reorder the terms: 1 + z = -2.915475947 Solving 1 + z = -2.915475947 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + z = -2.915475947 + -1 Combine like terms: 1 + -1 = 0 0 + z = -2.915475947 + -1 z = -2.915475947 + -1 Combine like terms: -2.915475947 + -1 = -3.915475947 z = -3.915475947 Simplifying z = -3.915475947Solution
The solution to the problem is based on the solutions from the subproblems. z = {1.915475947, -3.915475947}Solution
z = {1.915475947, -3.915475947}
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