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Simplifying -9x2 + 8x + 6 = 0 Reorder the terms: 6 + 8x + -9x2 = 0 Solving 6 + 8x + -9x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -0.6666666667 + -0.8888888889x + x2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -0.8888888889x + 0.6666666667 + x2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -0.8888888889x + x2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -0.8888888889x + x2 = 0 + 0.6666666667 -0.8888888889x + x2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -0.8888888889x + x2 = 0.6666666667 The x term is -0.8888888889x. Take half its coefficient (-0.4444444445). Square it (0.1975308642) and add it to both sides. Add '0.1975308642' to each side of the equation. -0.8888888889x + 0.1975308642 + x2 = 0.6666666667 + 0.1975308642 Reorder the terms: 0.1975308642 + -0.8888888889x + x2 = 0.6666666667 + 0.1975308642 Combine like terms: 0.6666666667 + 0.1975308642 = 0.8641975309 0.1975308642 + -0.8888888889x + x2 = 0.8641975309 Factor a perfect square on the left side: (x + -0.4444444445)(x + -0.4444444445) = 0.8641975309 Calculate the square root of the right side: 0.929622252 Break this problem into two subproblems by setting (x + -0.4444444445) equal to 0.929622252 and -0.929622252.Subproblem 1
x + -0.4444444445 = 0.929622252 Simplifying x + -0.4444444445 = 0.929622252 Reorder the terms: -0.4444444445 + x = 0.929622252 Solving -0.4444444445 + x = 0.929622252 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4444444445' to each side of the equation. -0.4444444445 + 0.4444444445 + x = 0.929622252 + 0.4444444445 Combine like terms: -0.4444444445 + 0.4444444445 = 0.0000000000 0.0000000000 + x = 0.929622252 + 0.4444444445 x = 0.929622252 + 0.4444444445 Combine like terms: 0.929622252 + 0.4444444445 = 1.3740666965 x = 1.3740666965 Simplifying x = 1.3740666965Subproblem 2
x + -0.4444444445 = -0.929622252 Simplifying x + -0.4444444445 = -0.929622252 Reorder the terms: -0.4444444445 + x = -0.929622252 Solving -0.4444444445 + x = -0.929622252 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4444444445' to each side of the equation. -0.4444444445 + 0.4444444445 + x = -0.929622252 + 0.4444444445 Combine like terms: -0.4444444445 + 0.4444444445 = 0.0000000000 0.0000000000 + x = -0.929622252 + 0.4444444445 x = -0.929622252 + 0.4444444445 Combine like terms: -0.929622252 + 0.4444444445 = -0.4851778075 x = -0.4851778075 Simplifying x = -0.4851778075Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.3740666965, -0.4851778075}
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