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Simplifying 25x2 + -24x + -9 = -7x2 + 12x + 18 Reorder the terms: -9 + -24x + 25x2 = -7x2 + 12x + 18 Reorder the terms: -9 + -24x + 25x2 = 18 + 12x + -7x2 Solving -9 + -24x + 25x2 = 18 + 12x + -7x2 Solving for variable 'x'. Reorder the terms: -9 + -18 + -24x + -12x + 25x2 + 7x2 = 18 + 12x + -7x2 + -18 + -12x + 7x2 Combine like terms: -9 + -18 = -27 -27 + -24x + -12x + 25x2 + 7x2 = 18 + 12x + -7x2 + -18 + -12x + 7x2 Combine like terms: -24x + -12x = -36x -27 + -36x + 25x2 + 7x2 = 18 + 12x + -7x2 + -18 + -12x + 7x2 Combine like terms: 25x2 + 7x2 = 32x2 -27 + -36x + 32x2 = 18 + 12x + -7x2 + -18 + -12x + 7x2 Reorder the terms: -27 + -36x + 32x2 = 18 + -18 + 12x + -12x + -7x2 + 7x2 Combine like terms: 18 + -18 = 0 -27 + -36x + 32x2 = 0 + 12x + -12x + -7x2 + 7x2 -27 + -36x + 32x2 = 12x + -12x + -7x2 + 7x2 Combine like terms: 12x + -12x = 0 -27 + -36x + 32x2 = 0 + -7x2 + 7x2 -27 + -36x + 32x2 = -7x2 + 7x2 Combine like terms: -7x2 + 7x2 = 0 -27 + -36x + 32x2 = 0 Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -0.84375 + -1.125x + x2 = 0 Move the constant term to the right: Add '0.84375' to each side of the equation. -0.84375 + -1.125x + 0.84375 + x2 = 0 + 0.84375 Reorder the terms: -0.84375 + 0.84375 + -1.125x + x2 = 0 + 0.84375 Combine like terms: -0.84375 + 0.84375 = 0.00000 0.00000 + -1.125x + x2 = 0 + 0.84375 -1.125x + x2 = 0 + 0.84375 Combine like terms: 0 + 0.84375 = 0.84375 -1.125x + x2 = 0.84375 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 = 0.84375 + 0.31640625 Reorder the terms: 0.31640625 + -1.125x + x2 = 0.84375 + 0.31640625 Combine like terms: 0.84375 + 0.31640625 = 1.16015625 0.31640625 + -1.125x + x2 = 1.16015625 Factor a perfect square on the left side: (x + -0.5625)(x + -0.5625) = 1.16015625 Calculate the square root of the right side: 1.077105496 Break this problem into two subproblems by setting (x + -0.5625) equal to 1.077105496 and -1.077105496.Subproblem 1
x + -0.5625 = 1.077105496 Simplifying x + -0.5625 = 1.077105496 Reorder the terms: -0.5625 + x = 1.077105496 Solving -0.5625 + x = 1.077105496 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 = 1.077105496 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + x = 1.077105496 + 0.5625 x = 1.077105496 + 0.5625 Combine like terms: 1.077105496 + 0.5625 = 1.639605496 x = 1.639605496 Simplifying x = 1.639605496Subproblem 2
x + -0.5625 = -1.077105496 Simplifying x + -0.5625 = -1.077105496 Reorder the terms: -0.5625 + x = -1.077105496 Solving -0.5625 + x = -1.077105496 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 = -1.077105496 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + x = -1.077105496 + 0.5625 x = -1.077105496 + 0.5625 Combine like terms: -1.077105496 + 0.5625 = -0.514605496 x = -0.514605496 Simplifying x = -0.514605496Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.639605496, -0.514605496}
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