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Simplifying 10x2 + 57x + 65 = 0 Reorder the terms: 65 + 57x + 10x2 = 0 Solving 65 + 57x + 10x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 6.5 + 5.7x + x2 = 0 Move the constant term to the right: Add '-6.5' to each side of the equation. 6.5 + 5.7x + -6.5 + x2 = 0 + -6.5 Reorder the terms: 6.5 + -6.5 + 5.7x + x2 = 0 + -6.5 Combine like terms: 6.5 + -6.5 = 0.0 0.0 + 5.7x + x2 = 0 + -6.5 5.7x + x2 = 0 + -6.5 Combine like terms: 0 + -6.5 = -6.5 5.7x + x2 = -6.5 The x term is 5.7x. Take half its coefficient (2.85). Square it (8.1225) and add it to both sides. Add '8.1225' to each side of the equation. 5.7x + 8.1225 + x2 = -6.5 + 8.1225 Reorder the terms: 8.1225 + 5.7x + x2 = -6.5 + 8.1225 Combine like terms: -6.5 + 8.1225 = 1.6225 8.1225 + 5.7x + x2 = 1.6225 Factor a perfect square on the left side: (x + 2.85)(x + 2.85) = 1.6225 Calculate the square root of the right side: 1.27377392 Break this problem into two subproblems by setting (x + 2.85) equal to 1.27377392 and -1.27377392.Subproblem 1
x + 2.85 = 1.27377392 Simplifying x + 2.85 = 1.27377392 Reorder the terms: 2.85 + x = 1.27377392 Solving 2.85 + x = 1.27377392 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.85' to each side of the equation. 2.85 + -2.85 + x = 1.27377392 + -2.85 Combine like terms: 2.85 + -2.85 = 0.00 0.00 + x = 1.27377392 + -2.85 x = 1.27377392 + -2.85 Combine like terms: 1.27377392 + -2.85 = -1.57622608 x = -1.57622608 Simplifying x = -1.57622608Subproblem 2
x + 2.85 = -1.27377392 Simplifying x + 2.85 = -1.27377392 Reorder the terms: 2.85 + x = -1.27377392 Solving 2.85 + x = -1.27377392 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.85' to each side of the equation. 2.85 + -2.85 + x = -1.27377392 + -2.85 Combine like terms: 2.85 + -2.85 = 0.00 0.00 + x = -1.27377392 + -2.85 x = -1.27377392 + -2.85 Combine like terms: -1.27377392 + -2.85 = -4.12377392 x = -4.12377392 Simplifying x = -4.12377392Solution
The solution to the problem is based on the solutions from the subproblems. x = {-1.57622608, -4.12377392}
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