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Simplifying 30z2 + -35z2 + -24z + 28 = 0 Reorder the terms: 28 + -24z + 30z2 + -35z2 = 0 Combine like terms: 30z2 + -35z2 = -5z2 28 + -24z + -5z2 = 0 Solving 28 + -24z + -5z2 = 0 Solving for variable 'z'. Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -5.6 + 4.8z + z2 = 0 Move the constant term to the right: Add '5.6' to each side of the equation. -5.6 + 4.8z + 5.6 + z2 = 0 + 5.6 Reorder the terms: -5.6 + 5.6 + 4.8z + z2 = 0 + 5.6 Combine like terms: -5.6 + 5.6 = 0.0 0.0 + 4.8z + z2 = 0 + 5.6 4.8z + z2 = 0 + 5.6 Combine like terms: 0 + 5.6 = 5.6 4.8z + z2 = 5.6 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 = 5.6 + 5.76 Reorder the terms: 5.76 + 4.8z + z2 = 5.6 + 5.76 Combine like terms: 5.6 + 5.76 = 11.36 5.76 + 4.8z + z2 = 11.36 Factor a perfect square on the left side: (z + 2.4)(z + 2.4) = 11.36 Calculate the square root of the right side: 3.370459909 Break this problem into two subproblems by setting (z + 2.4) equal to 3.370459909 and -3.370459909.Subproblem 1
z + 2.4 = 3.370459909 Simplifying z + 2.4 = 3.370459909 Reorder the terms: 2.4 + z = 3.370459909 Solving 2.4 + z = 3.370459909 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 = 3.370459909 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + z = 3.370459909 + -2.4 z = 3.370459909 + -2.4 Combine like terms: 3.370459909 + -2.4 = 0.970459909 z = 0.970459909 Simplifying z = 0.970459909Subproblem 2
z + 2.4 = -3.370459909 Simplifying z + 2.4 = -3.370459909 Reorder the terms: 2.4 + z = -3.370459909 Solving 2.4 + z = -3.370459909 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 = -3.370459909 + -2.4 Combine like terms: 2.4 + -2.4 = 0.0 0.0 + z = -3.370459909 + -2.4 z = -3.370459909 + -2.4 Combine like terms: -3.370459909 + -2.4 = -5.770459909 z = -5.770459909 Simplifying z = -5.770459909Solution
The solution to the problem is based on the solutions from the subproblems. z = {0.970459909, -5.770459909}
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