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Simplifying (z + -5)(z + -4) = 10 Reorder the terms: (-5 + z)(z + -4) = 10 Reorder the terms: (-5 + z)(-4 + z) = 10 Multiply (-5 + z) * (-4 + z) (-5(-4 + z) + z(-4 + z)) = 10 ((-4 * -5 + z * -5) + z(-4 + z)) = 10 ((20 + -5z) + z(-4 + z)) = 10 (20 + -5z + (-4 * z + z * z)) = 10 (20 + -5z + (-4z + z2)) = 10 Combine like terms: -5z + -4z = -9z (20 + -9z + z2) = 10 Solving 20 + -9z + z2 = 10 Solving for variable 'z'. Reorder the terms: 20 + -10 + -9z + z2 = 10 + -10 Combine like terms: 20 + -10 = 10 10 + -9z + z2 = 10 + -10 Combine like terms: 10 + -10 = 0 10 + -9z + z2 = 0 Begin completing the square. Move the constant term to the right: Add '-10' to each side of the equation. 10 + -9z + -10 + z2 = 0 + -10 Reorder the terms: 10 + -10 + -9z + z2 = 0 + -10 Combine like terms: 10 + -10 = 0 0 + -9z + z2 = 0 + -10 -9z + z2 = 0 + -10 Combine like terms: 0 + -10 = -10 -9z + z2 = -10 The z term is -9z. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9z + 20.25 + z2 = -10 + 20.25 Reorder the terms: 20.25 + -9z + z2 = -10 + 20.25 Combine like terms: -10 + 20.25 = 10.25 20.25 + -9z + z2 = 10.25 Factor a perfect square on the left side: (z + -4.5)(z + -4.5) = 10.25 Calculate the square root of the right side: 3.201562119 Break this problem into two subproblems by setting (z + -4.5) equal to 3.201562119 and -3.201562119.Subproblem 1
z + -4.5 = 3.201562119 Simplifying z + -4.5 = 3.201562119 Reorder the terms: -4.5 + z = 3.201562119 Solving -4.5 + z = 3.201562119 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + z = 3.201562119 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + z = 3.201562119 + 4.5 z = 3.201562119 + 4.5 Combine like terms: 3.201562119 + 4.5 = 7.701562119 z = 7.701562119 Simplifying z = 7.701562119Subproblem 2
z + -4.5 = -3.201562119 Simplifying z + -4.5 = -3.201562119 Reorder the terms: -4.5 + z = -3.201562119 Solving -4.5 + z = -3.201562119 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + z = -3.201562119 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + z = -3.201562119 + 4.5 z = -3.201562119 + 4.5 Combine like terms: -3.201562119 + 4.5 = 1.298437881 z = 1.298437881 Simplifying z = 1.298437881Solution
The solution to the problem is based on the solutions from the subproblems. z = {7.701562119, 1.298437881}
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