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Simplifying 6.5(x)(x) + 7x + -8.5 = 0 Multiply x * x 6.5x2 + 7x + -8.5 = 0 Reorder the terms: -8.5 + 7x + 6.5x2 = 0 Solving -8.5 + 7x + 6.5x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 6.5 the coefficient of the squared term: Divide each side by '6.5'. -1.307692308 + 1.076923077x + x2 = 0 Move the constant term to the right: Add '1.307692308' to each side of the equation. -1.307692308 + 1.076923077x + 1.307692308 + x2 = 0 + 1.307692308 Reorder the terms: -1.307692308 + 1.307692308 + 1.076923077x + x2 = 0 + 1.307692308 Combine like terms: -1.307692308 + 1.307692308 = 0.000000000 0.000000000 + 1.076923077x + x2 = 0 + 1.307692308 1.076923077x + x2 = 0 + 1.307692308 Combine like terms: 0 + 1.307692308 = 1.307692308 1.076923077x + x2 = 1.307692308 The x term is 1.076923077x. Take half its coefficient (0.5384615385). Square it (0.2899408284) and add it to both sides. Add '0.2899408284' to each side of the equation. 1.076923077x + 0.2899408284 + x2 = 1.307692308 + 0.2899408284 Reorder the terms: 0.2899408284 + 1.076923077x + x2 = 1.307692308 + 0.2899408284 Combine like terms: 1.307692308 + 0.2899408284 = 1.5976331364 0.2899408284 + 1.076923077x + x2 = 1.5976331364 Factor a perfect square on the left side: (x + 0.5384615385)(x + 0.5384615385) = 1.5976331364 Calculate the square root of the right side: 1.263975133 Break this problem into two subproblems by setting (x + 0.5384615385) equal to 1.263975133 and -1.263975133.Subproblem 1
x + 0.5384615385 = 1.263975133 Simplifying x + 0.5384615385 = 1.263975133 Reorder the terms: 0.5384615385 + x = 1.263975133 Solving 0.5384615385 + x = 1.263975133 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5384615385' to each side of the equation. 0.5384615385 + -0.5384615385 + x = 1.263975133 + -0.5384615385 Combine like terms: 0.5384615385 + -0.5384615385 = 0.0000000000 0.0000000000 + x = 1.263975133 + -0.5384615385 x = 1.263975133 + -0.5384615385 Combine like terms: 1.263975133 + -0.5384615385 = 0.7255135945 x = 0.7255135945 Simplifying x = 0.7255135945Subproblem 2
x + 0.5384615385 = -1.263975133 Simplifying x + 0.5384615385 = -1.263975133 Reorder the terms: 0.5384615385 + x = -1.263975133 Solving 0.5384615385 + x = -1.263975133 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5384615385' to each side of the equation. 0.5384615385 + -0.5384615385 + x = -1.263975133 + -0.5384615385 Combine like terms: 0.5384615385 + -0.5384615385 = 0.0000000000 0.0000000000 + x = -1.263975133 + -0.5384615385 x = -1.263975133 + -0.5384615385 Combine like terms: -1.263975133 + -0.5384615385 = -1.8024366715 x = -1.8024366715 Simplifying x = -1.8024366715Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.7255135945, -1.8024366715}
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