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Simplifying 0.75x2 + 2x = 9 Reorder the terms: 2x + 0.75x2 = 9 Solving 2x + 0.75x2 = 9 Solving for variable 'x'. Reorder the terms: -9 + 2x + 0.75x2 = 9 + -9 Combine like terms: 9 + -9 = 0 -9 + 2x + 0.75x2 = 0 Begin completing the square. Divide all terms by 0.75 the coefficient of the squared term: Divide each side by '0.75'. -12 + 2.666666667x + x2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + 2.666666667x + 12 + x2 = 0 + 12 Reorder the terms: -12 + 12 + 2.666666667x + x2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 2.666666667x + x2 = 0 + 12 2.666666667x + x2 = 0 + 12 Combine like terms: 0 + 12 = 12 2.666666667x + x2 = 12 The x term is 2.666666667x. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667x + 1.777777780 + x2 = 12 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667x + x2 = 12 + 1.777777780 Combine like terms: 12 + 1.777777780 = 13.77777778 1.777777780 + 2.666666667x + x2 = 13.77777778 Factor a perfect square on the left side: (x + 1.333333334)(x + 1.333333334) = 13.77777778 Calculate the square root of the right side: 3.711842909 Break this problem into two subproblems by setting (x + 1.333333334) equal to 3.711842909 and -3.711842909.Subproblem 1
x + 1.333333334 = 3.711842909 Simplifying x + 1.333333334 = 3.711842909 Reorder the terms: 1.333333334 + x = 3.711842909 Solving 1.333333334 + x = 3.711842909 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = 3.711842909 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = 3.711842909 + -1.333333334 x = 3.711842909 + -1.333333334 Combine like terms: 3.711842909 + -1.333333334 = 2.378509575 x = 2.378509575 Simplifying x = 2.378509575Subproblem 2
x + 1.333333334 = -3.711842909 Simplifying x + 1.333333334 = -3.711842909 Reorder the terms: 1.333333334 + x = -3.711842909 Solving 1.333333334 + x = -3.711842909 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + x = -3.711842909 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + x = -3.711842909 + -1.333333334 x = -3.711842909 + -1.333333334 Combine like terms: -3.711842909 + -1.333333334 = -5.045176243 x = -5.045176243 Simplifying x = -5.045176243Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.378509575, -5.045176243}
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