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Simplifying 5x2 + 7x + -6 = x2 + 2x + -3.5 Reorder the terms: -6 + 7x + 5x2 = x2 + 2x + -3.5 Reorder the terms: -6 + 7x + 5x2 = -3.5 + 2x + x2 Solving -6 + 7x + 5x2 = -3.5 + 2x + x2 Solving for variable 'x'. Reorder the terms: -6 + 3.5 + 7x + -2x + 5x2 + -1x2 = -3.5 + 2x + x2 + 3.5 + -2x + -1x2 Combine like terms: -6 + 3.5 = -2.5 -2.5 + 7x + -2x + 5x2 + -1x2 = -3.5 + 2x + x2 + 3.5 + -2x + -1x2 Combine like terms: 7x + -2x = 5x -2.5 + 5x + 5x2 + -1x2 = -3.5 + 2x + x2 + 3.5 + -2x + -1x2 Combine like terms: 5x2 + -1x2 = 4x2 -2.5 + 5x + 4x2 = -3.5 + 2x + x2 + 3.5 + -2x + -1x2 Reorder the terms: -2.5 + 5x + 4x2 = -3.5 + 3.5 + 2x + -2x + x2 + -1x2 Combine like terms: -3.5 + 3.5 = 0.0 -2.5 + 5x + 4x2 = 0.0 + 2x + -2x + x2 + -1x2 -2.5 + 5x + 4x2 = 2x + -2x + x2 + -1x2 Combine like terms: 2x + -2x = 0 -2.5 + 5x + 4x2 = 0 + x2 + -1x2 -2.5 + 5x + 4x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -2.5 + 5x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.625 + 1.25x + x2 = 0 Move the constant term to the right: Add '0.625' to each side of the equation. -0.625 + 1.25x + 0.625 + x2 = 0 + 0.625 Reorder the terms: -0.625 + 0.625 + 1.25x + x2 = 0 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + 1.25x + x2 = 0 + 0.625 1.25x + x2 = 0 + 0.625 Combine like terms: 0 + 0.625 = 0.625 1.25x + x2 = 0.625 The x term is 1.25x. Take half its coefficient (0.625). Square it (0.390625) and add it to both sides. Add '0.390625' to each side of the equation. 1.25x + 0.390625 + x2 = 0.625 + 0.390625 Reorder the terms: 0.390625 + 1.25x + x2 = 0.625 + 0.390625 Combine like terms: 0.625 + 0.390625 = 1.015625 0.390625 + 1.25x + x2 = 1.015625 Factor a perfect square on the left side: (x + 0.625)(x + 0.625) = 1.015625 Calculate the square root of the right side: 1.007782219 Break this problem into two subproblems by setting (x + 0.625) equal to 1.007782219 and -1.007782219.Subproblem 1
x + 0.625 = 1.007782219 Simplifying x + 0.625 = 1.007782219 Reorder the terms: 0.625 + x = 1.007782219 Solving 0.625 + x = 1.007782219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = 1.007782219 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = 1.007782219 + -0.625 x = 1.007782219 + -0.625 Combine like terms: 1.007782219 + -0.625 = 0.382782219 x = 0.382782219 Simplifying x = 0.382782219Subproblem 2
x + 0.625 = -1.007782219 Simplifying x + 0.625 = -1.007782219 Reorder the terms: 0.625 + x = -1.007782219 Solving 0.625 + x = -1.007782219 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.625' to each side of the equation. 0.625 + -0.625 + x = -1.007782219 + -0.625 Combine like terms: 0.625 + -0.625 = 0.000 0.000 + x = -1.007782219 + -0.625 x = -1.007782219 + -0.625 Combine like terms: -1.007782219 + -0.625 = -1.632782219 x = -1.632782219 Simplifying x = -1.632782219Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.382782219, -1.632782219}
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