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Simplifying 27x2 + 32x = 52 Reorder the terms: 32x + 27x2 = 52 Solving 32x + 27x2 = 52 Solving for variable 'x'. Reorder the terms: -52 + 32x + 27x2 = 52 + -52 Combine like terms: 52 + -52 = 0 -52 + 32x + 27x2 = 0 Begin completing the square. Divide all terms by 27 the coefficient of the squared term: Divide each side by '27'. -1.925925926 + 1.185185185x + x2 = 0 Move the constant term to the right: Add '1.925925926' to each side of the equation. -1.925925926 + 1.185185185x + 1.925925926 + x2 = 0 + 1.925925926 Reorder the terms: -1.925925926 + 1.925925926 + 1.185185185x + x2 = 0 + 1.925925926 Combine like terms: -1.925925926 + 1.925925926 = 0.000000000 0.000000000 + 1.185185185x + x2 = 0 + 1.925925926 1.185185185x + x2 = 0 + 1.925925926 Combine like terms: 0 + 1.925925926 = 1.925925926 1.185185185x + x2 = 1.925925926 The x term is 1.185185185x. Take half its coefficient (0.5925925925). Square it (0.3511659807) and add it to both sides. Add '0.3511659807' to each side of the equation. 1.185185185x + 0.3511659807 + x2 = 1.925925926 + 0.3511659807 Reorder the terms: 0.3511659807 + 1.185185185x + x2 = 1.925925926 + 0.3511659807 Combine like terms: 1.925925926 + 0.3511659807 = 2.2770919067 0.3511659807 + 1.185185185x + x2 = 2.2770919067 Factor a perfect square on the left side: (x + 0.5925925925)(x + 0.5925925925) = 2.2770919067 Calculate the square root of the right side: 1.509003614 Break this problem into two subproblems by setting (x + 0.5925925925) equal to 1.509003614 and -1.509003614.Subproblem 1
x + 0.5925925925 = 1.509003614 Simplifying x + 0.5925925925 = 1.509003614 Reorder the terms: 0.5925925925 + x = 1.509003614 Solving 0.5925925925 + x = 1.509003614 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5925925925' to each side of the equation. 0.5925925925 + -0.5925925925 + x = 1.509003614 + -0.5925925925 Combine like terms: 0.5925925925 + -0.5925925925 = 0.0000000000 0.0000000000 + x = 1.509003614 + -0.5925925925 x = 1.509003614 + -0.5925925925 Combine like terms: 1.509003614 + -0.5925925925 = 0.9164110215 x = 0.9164110215 Simplifying x = 0.9164110215Subproblem 2
x + 0.5925925925 = -1.509003614 Simplifying x + 0.5925925925 = -1.509003614 Reorder the terms: 0.5925925925 + x = -1.509003614 Solving 0.5925925925 + x = -1.509003614 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5925925925' to each side of the equation. 0.5925925925 + -0.5925925925 + x = -1.509003614 + -0.5925925925 Combine like terms: 0.5925925925 + -0.5925925925 = 0.0000000000 0.0000000000 + x = -1.509003614 + -0.5925925925 x = -1.509003614 + -0.5925925925 Combine like terms: -1.509003614 + -0.5925925925 = -2.1015962065 x = -2.1015962065 Simplifying x = -2.1015962065Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.9164110215, -2.1015962065}
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