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Simplifying 49z2 + 42z = 19 Reorder the terms: 42z + 49z2 = 19 Solving 42z + 49z2 = 19 Solving for variable 'z'. Reorder the terms: -19 + 42z + 49z2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + 42z + 49z2 = 0 Begin completing the square. Divide all terms by 49 the coefficient of the squared term: Divide each side by '49'. -0.387755102 + 0.8571428571z + z2 = 0 Move the constant term to the right: Add '0.387755102' to each side of the equation. -0.387755102 + 0.8571428571z + 0.387755102 + z2 = 0 + 0.387755102 Reorder the terms: -0.387755102 + 0.387755102 + 0.8571428571z + z2 = 0 + 0.387755102 Combine like terms: -0.387755102 + 0.387755102 = 0.000000000 0.000000000 + 0.8571428571z + z2 = 0 + 0.387755102 0.8571428571z + z2 = 0 + 0.387755102 Combine like terms: 0 + 0.387755102 = 0.387755102 0.8571428571z + z2 = 0.387755102 The z term is 0.8571428571z. Take half its coefficient (0.4285714286). Square it (0.1836734694) and add it to both sides. Add '0.1836734694' to each side of the equation. 0.8571428571z + 0.1836734694 + z2 = 0.387755102 + 0.1836734694 Reorder the terms: 0.1836734694 + 0.8571428571z + z2 = 0.387755102 + 0.1836734694 Combine like terms: 0.387755102 + 0.1836734694 = 0.5714285714 0.1836734694 + 0.8571428571z + z2 = 0.5714285714 Factor a perfect square on the left side: (z + 0.4285714286)(z + 0.4285714286) = 0.5714285714 Calculate the square root of the right side: 0.755928946 Break this problem into two subproblems by setting (z + 0.4285714286) equal to 0.755928946 and -0.755928946.Subproblem 1
z + 0.4285714286 = 0.755928946 Simplifying z + 0.4285714286 = 0.755928946 Reorder the terms: 0.4285714286 + z = 0.755928946 Solving 0.4285714286 + z = 0.755928946 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + z = 0.755928946 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + z = 0.755928946 + -0.4285714286 z = 0.755928946 + -0.4285714286 Combine like terms: 0.755928946 + -0.4285714286 = 0.3273575174 z = 0.3273575174 Simplifying z = 0.3273575174Subproblem 2
z + 0.4285714286 = -0.755928946 Simplifying z + 0.4285714286 = -0.755928946 Reorder the terms: 0.4285714286 + z = -0.755928946 Solving 0.4285714286 + z = -0.755928946 Solving for variable 'z'. Move all terms containing z to the left, all other terms to the right. Add '-0.4285714286' to each side of the equation. 0.4285714286 + -0.4285714286 + z = -0.755928946 + -0.4285714286 Combine like terms: 0.4285714286 + -0.4285714286 = 0.0000000000 0.0000000000 + z = -0.755928946 + -0.4285714286 z = -0.755928946 + -0.4285714286 Combine like terms: -0.755928946 + -0.4285714286 = -1.1845003746 z = -1.1845003746 Simplifying z = -1.1845003746Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.3273575174, -1.1845003746}
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