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Simplifying 0.75x2 + 5x + -15 = 0 Reorder the terms: -15 + 5x + 0.75x2 = 0 Solving -15 + 5x + 0.75x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 0.75 the coefficient of the squared term: Divide each side by '0.75'. -20 + 6.666666667x + x2 = 0 Move the constant term to the right: Add '20' to each side of the equation. -20 + 6.666666667x + 20 + x2 = 0 + 20 Reorder the terms: -20 + 20 + 6.666666667x + x2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + 6.666666667x + x2 = 0 + 20 6.666666667x + x2 = 0 + 20 Combine like terms: 0 + 20 = 20 6.666666667x + x2 = 20 The x term is 6.666666667x. Take half its coefficient (3.333333334). Square it (11.11111112) and add it to both sides. Add '11.11111112' to each side of the equation. 6.666666667x + 11.11111112 + x2 = 20 + 11.11111112 Reorder the terms: 11.11111112 + 6.666666667x + x2 = 20 + 11.11111112 Combine like terms: 20 + 11.11111112 = 31.11111112 11.11111112 + 6.666666667x + x2 = 31.11111112 Factor a perfect square on the left side: (x + 3.333333334)(x + 3.333333334) = 31.11111112 Calculate the square root of the right side: 5.577733511 Break this problem into two subproblems by setting (x + 3.333333334) equal to 5.577733511 and -5.577733511.Subproblem 1
x + 3.333333334 = 5.577733511 Simplifying x + 3.333333334 = 5.577733511 Reorder the terms: 3.333333334 + x = 5.577733511 Solving 3.333333334 + x = 5.577733511 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.333333334' to each side of the equation. 3.333333334 + -3.333333334 + x = 5.577733511 + -3.333333334 Combine like terms: 3.333333334 + -3.333333334 = 0.000000000 0.000000000 + x = 5.577733511 + -3.333333334 x = 5.577733511 + -3.333333334 Combine like terms: 5.577733511 + -3.333333334 = 2.244400177 x = 2.244400177 Simplifying x = 2.244400177Subproblem 2
x + 3.333333334 = -5.577733511 Simplifying x + 3.333333334 = -5.577733511 Reorder the terms: 3.333333334 + x = -5.577733511 Solving 3.333333334 + x = -5.577733511 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.333333334' to each side of the equation. 3.333333334 + -3.333333334 + x = -5.577733511 + -3.333333334 Combine like terms: 3.333333334 + -3.333333334 = 0.000000000 0.000000000 + x = -5.577733511 + -3.333333334 x = -5.577733511 + -3.333333334 Combine like terms: -5.577733511 + -3.333333334 = -8.911066845 x = -8.911066845 Simplifying x = -8.911066845Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.244400177, -8.911066845}
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