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Simplifying 5z(z + -4) = 4(z + -3) Reorder the terms: 5z(-4 + z) = 4(z + -3) (-4 * 5z + z * 5z) = 4(z + -3) (-20z + 5z2) = 4(z + -3) Reorder the terms: -20z + 5z2 = 4(-3 + z) -20z + 5z2 = (-3 * 4 + z * 4) -20z + 5z2 = (-12 + 4z) Solving -20z + 5z2 = -12 + 4z Solving for variable 'z'. Reorder the terms: 12 + -20z + -4z + 5z2 = -12 + 4z + 12 + -4z Combine like terms: -20z + -4z = -24z 12 + -24z + 5z2 = -12 + 4z + 12 + -4z Reorder the terms: 12 + -24z + 5z2 = -12 + 12 + 4z + -4z Combine like terms: -12 + 12 = 0 12 + -24z + 5z2 = 0 + 4z + -4z 12 + -24z + 5z2 = 4z + -4z Combine like terms: 4z + -4z = 0 12 + -24z + 5z2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 2.4 + -4.8z + z2 = 0 Move the constant term to the right: Add '-2.4' to each side of the equation. 2.4 + -4.8z + -2.4 + z2 = 0 + -2.4 Reorder the terms: 2.4 + -2.4 + -4.8z + z2 = 0 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + -4.8z + z2 = 0 + -2.4 -4.8z + z2 = 0 + -2.4 Combine like terms: 0 + -2.4 = -2.4 -4.8z + z2 = -2.4 The z term is -4.8z. Take half its coefficient (-2.4). Square it (5.76) and add it to both sides. Add '5.76' to each side of the equation. -4.8z + 5.76 + z2 = -2.4 + 5.76 Reorder the terms: 5.76 + -4.8z + z2 = -2.4 + 5.76 Combine like terms: -2.4 + 5.76 = 3.36 5.76 + -4.8z + z2 = 3.36 Factor a perfect square on the left side: (z + -2.4)(z + -2.4) = 3.36 Calculate the square root of the right side: 1.833030278 Break this problem into two subproblems by setting (z + -2.4) equal to 1.833030278 and -1.833030278.Subproblem 1
z + -2.4 = 1.833030278 Simplifying z + -2.4 = 1.833030278 Reorder the terms: -2.4 + z = 1.833030278 Solving -2.4 + z = 1.833030278 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.4' to each side of the equation. -2.4 + 2.4 + z = 1.833030278 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + z = 1.833030278 + 2.4 z = 1.833030278 + 2.4 Combine like terms: 1.833030278 + 2.4 = 4.233030278 z = 4.233030278 Simplifying z = 4.233030278Subproblem 2
z + -2.4 = -1.833030278 Simplifying z + -2.4 = -1.833030278 Reorder the terms: -2.4 + z = -1.833030278 Solving -2.4 + z = -1.833030278 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '2.4' to each side of the equation. -2.4 + 2.4 + z = -1.833030278 + 2.4 Combine like terms: -2.4 + 2.4 = 0.0 0.0 + z = -1.833030278 + 2.4 z = -1.833030278 + 2.4 Combine like terms: -1.833030278 + 2.4 = 0.566969722 z = 0.566969722 Simplifying z = 0.566969722Solution
The solution to the problem is based on the solutions from the subproblems. z = {4.233030278, 0.566969722}
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