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Simplifying 0 = 7.8x2 + -7.8x + 1 Reorder the terms: 0 = 1 + -7.8x + 7.8x2 Solving 0 = 1 + -7.8x + 7.8x2 Solving for variable 'x'. Combine like terms: 0 + -1 = -1 -1 + 7.8x + -7.8x2 = 1 + -7.8x + 7.8x2 + -1 + 7.8x + -7.8x2 Reorder the terms: -1 + 7.8x + -7.8x2 = 1 + -1 + -7.8x + 7.8x + 7.8x2 + -7.8x2 Combine like terms: 1 + -1 = 0 -1 + 7.8x + -7.8x2 = 0 + -7.8x + 7.8x + 7.8x2 + -7.8x2 -1 + 7.8x + -7.8x2 = -7.8x + 7.8x + 7.8x2 + -7.8x2 Combine like terms: -7.8x + 7.8x = 0.0 -1 + 7.8x + -7.8x2 = 0.0 + 7.8x2 + -7.8x2 -1 + 7.8x + -7.8x2 = 7.8x2 + -7.8x2 Combine like terms: 7.8x2 + -7.8x2 = 0.0 -1 + 7.8x + -7.8x2 = 0.0 Begin completing the square. Divide all terms by -7.8 the coefficient of the squared term: Divide each side by '-7.8'. 0.1282051282 + -1x + x2 = 0 Move the constant term to the right: Add '-0.1282051282' to each side of the equation. 0.1282051282 + -1x + -0.1282051282 + x2 = 0 + -0.1282051282 Reorder the terms: 0.1282051282 + -0.1282051282 + -1x + x2 = 0 + -0.1282051282 Combine like terms: 0.1282051282 + -0.1282051282 = 0.0000000000 0.0000000000 + -1x + x2 = 0 + -0.1282051282 -1x + x2 = 0 + -0.1282051282 Combine like terms: 0 + -0.1282051282 = -0.1282051282 -1x + x2 = -0.1282051282 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = -0.1282051282 + 0.25 Reorder the terms: 0.25 + -1x + x2 = -0.1282051282 + 0.25 Combine like terms: -0.1282051282 + 0.25 = 0.1217948718 0.25 + -1x + x2 = 0.1217948718 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 0.1217948718 Calculate the square root of the right side: 0.34899122 Break this problem into two subproblems by setting (x + -0.5) equal to 0.34899122 and -0.34899122.Subproblem 1
x + -0.5 = 0.34899122 Simplifying x + -0.5 = 0.34899122 Reorder the terms: -0.5 + x = 0.34899122 Solving -0.5 + x = 0.34899122 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 0.34899122 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.34899122 + 0.5 x = 0.34899122 + 0.5 Combine like terms: 0.34899122 + 0.5 = 0.84899122 x = 0.84899122 Simplifying x = 0.84899122Subproblem 2
x + -0.5 = -0.34899122 Simplifying x + -0.5 = -0.34899122 Reorder the terms: -0.5 + x = -0.34899122 Solving -0.5 + x = -0.34899122 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -0.34899122 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.34899122 + 0.5 x = -0.34899122 + 0.5 Combine like terms: -0.34899122 + 0.5 = 0.15100878 x = 0.15100878 Simplifying x = 0.15100878Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.84899122, 0.15100878}
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