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Simplifying 25x2 + -24x + -9 = -7x2 + 12x Reorder the terms: -9 + -24x + 25x2 = -7x2 + 12x Reorder the terms: -9 + -24x + 25x2 = 12x + -7x2 Solving -9 + -24x + 25x2 = 12x + -7x2 Solving for variable 'x'. Reorder the terms: -9 + -24x + -12x + 25x2 + 7x2 = 12x + -7x2 + -12x + 7x2 Combine like terms: -24x + -12x = -36x -9 + -36x + 25x2 + 7x2 = 12x + -7x2 + -12x + 7x2 Combine like terms: 25x2 + 7x2 = 32x2 -9 + -36x + 32x2 = 12x + -7x2 + -12x + 7x2 Reorder the terms: -9 + -36x + 32x2 = 12x + -12x + -7x2 + 7x2 Combine like terms: 12x + -12x = 0 -9 + -36x + 32x2 = 0 + -7x2 + 7x2 -9 + -36x + 32x2 = -7x2 + 7x2 Combine like terms: -7x2 + 7x2 = 0 -9 + -36x + 32x2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -0.28125 + -1.125x + x2 = 0 Move the constant term to the right: Add '0.28125' to each side of the equation. -0.28125 + -1.125x + 0.28125 + x2 = 0 + 0.28125 Reorder the terms: -0.28125 + 0.28125 + -1.125x + x2 = 0 + 0.28125 Combine like terms: -0.28125 + 0.28125 = 0.00000 0.00000 + -1.125x + x2 = 0 + 0.28125 -1.125x + x2 = 0 + 0.28125 Combine like terms: 0 + 0.28125 = 0.28125 -1.125x + x2 = 0.28125 The x term is -1.125x. Take half its coefficient (-0.5625). Square it (0.31640625) and add it to both sides. Add '0.31640625' to each side of the equation. -1.125x + 0.31640625 + x2 = 0.28125 + 0.31640625 Reorder the terms: 0.31640625 + -1.125x + x2 = 0.28125 + 0.31640625 Combine like terms: 0.28125 + 0.31640625 = 0.59765625 0.31640625 + -1.125x + x2 = 0.59765625 Factor a perfect square on the left side: (x + -0.5625)(x + -0.5625) = 0.59765625 Calculate the square root of the right side: 0.773082305 Break this problem into two subproblems by setting (x + -0.5625) equal to 0.773082305 and -0.773082305.Subproblem 1
x + -0.5625 = 0.773082305 Simplifying x + -0.5625 = 0.773082305 Reorder the terms: -0.5625 + x = 0.773082305 Solving -0.5625 + x = 0.773082305 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + x = 0.773082305 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + x = 0.773082305 + 0.5625 x = 0.773082305 + 0.5625 Combine like terms: 0.773082305 + 0.5625 = 1.335582305 x = 1.335582305 Simplifying x = 1.335582305Subproblem 2
x + -0.5625 = -0.773082305 Simplifying x + -0.5625 = -0.773082305 Reorder the terms: -0.5625 + x = -0.773082305 Solving -0.5625 + x = -0.773082305 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5625' to each side of the equation. -0.5625 + 0.5625 + x = -0.773082305 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + x = -0.773082305 + 0.5625 x = -0.773082305 + 0.5625 Combine like terms: -0.773082305 + 0.5625 = -0.210582305 x = -0.210582305 Simplifying x = -0.210582305Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.335582305, -0.210582305}
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