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Simplifying (3x2 + -5x + 7) + -1(4x2 + -3x) = 0 Reorder the terms: (7 + -5x + 3x2) + -1(4x2 + -3x) = 0 Remove parenthesis around (7 + -5x + 3x2) 7 + -5x + 3x2 + -1(4x2 + -3x) = 0 Reorder the terms: 7 + -5x + 3x2 + -1(-3x + 4x2) = 0 7 + -5x + 3x2 + (-3x * -1 + 4x2 * -1) = 0 7 + -5x + 3x2 + (3x + -4x2) = 0 Reorder the terms: 7 + -5x + 3x + 3x2 + -4x2 = 0 Combine like terms: -5x + 3x = -2x 7 + -2x + 3x2 + -4x2 = 0 Combine like terms: 3x2 + -4x2 = -1x2 7 + -2x + -1x2 = 0 Solving 7 + -2x + -1x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -7 + 2x + x2 = 0 Move the constant term to the right: Add '7' to each side of the equation. -7 + 2x + 7 + x2 = 0 + 7 Reorder the terms: -7 + 7 + 2x + x2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 2x + x2 = 0 + 7 2x + x2 = 0 + 7 Combine like terms: 0 + 7 = 7 2x + x2 = 7 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 7 + 1 Reorder the terms: 1 + 2x + x2 = 7 + 1 Combine like terms: 7 + 1 = 8 1 + 2x + x2 = 8 Factor a perfect square on the left side: (x + 1)(x + 1) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (x + 1) equal to 2.828427125 and -2.828427125.Subproblem 1
x + 1 = 2.828427125 Simplifying x + 1 = 2.828427125 Reorder the terms: 1 + x = 2.828427125 Solving 1 + x = 2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 2.828427125 + -1 Combine like terms: 1 + -1 = 0 0 + x = 2.828427125 + -1 x = 2.828427125 + -1 Combine like terms: 2.828427125 + -1 = 1.828427125 x = 1.828427125 Simplifying x = 1.828427125Subproblem 2
x + 1 = -2.828427125 Simplifying x + 1 = -2.828427125 Reorder the terms: 1 + x = -2.828427125 Solving 1 + x = -2.828427125 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -2.828427125 + -1 Combine like terms: 1 + -1 = 0 0 + x = -2.828427125 + -1 x = -2.828427125 + -1 Combine like terms: -2.828427125 + -1 = -3.828427125 x = -3.828427125 Simplifying x = -3.828427125Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.828427125, -3.828427125}
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