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Simplifying 3x + -4x2 + 1 = 6 + -7x Reorder the terms: 1 + 3x + -4x2 = 6 + -7x Solving 1 + 3x + -4x2 = 6 + -7x Solving for variable 'x'. Reorder the terms: 1 + -6 + 3x + 7x + -4x2 = 6 + -7x + -6 + 7x Combine like terms: 1 + -6 = -5 -5 + 3x + 7x + -4x2 = 6 + -7x + -6 + 7x Combine like terms: 3x + 7x = 10x -5 + 10x + -4x2 = 6 + -7x + -6 + 7x Reorder the terms: -5 + 10x + -4x2 = 6 + -6 + -7x + 7x Combine like terms: 6 + -6 = 0 -5 + 10x + -4x2 = 0 + -7x + 7x -5 + 10x + -4x2 = -7x + 7x Combine like terms: -7x + 7x = 0 -5 + 10x + -4x2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. 1.25 + -2.5x + x2 = 0 Move the constant term to the right: Add '-1.25' to each side of the equation. 1.25 + -2.5x + -1.25 + x2 = 0 + -1.25 Reorder the terms: 1.25 + -1.25 + -2.5x + x2 = 0 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + -2.5x + x2 = 0 + -1.25 -2.5x + x2 = 0 + -1.25 Combine like terms: 0 + -1.25 = -1.25 -2.5x + x2 = -1.25 The x term is -2.5x. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5x + 1.5625 + x2 = -1.25 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = -1.25 + 1.5625 Combine like terms: -1.25 + 1.5625 = 0.3125 1.5625 + -2.5x + x2 = 0.3125 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (x + -1.25) equal to 0.559016994 and -0.559016994.Subproblem 1
x + -1.25 = 0.559016994 Simplifying x + -1.25 = 0.559016994 Reorder the terms: -1.25 + x = 0.559016994 Solving -1.25 + x = 0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 0.559016994 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 0.559016994 + 1.25 x = 0.559016994 + 1.25 Combine like terms: 0.559016994 + 1.25 = 1.809016994 x = 1.809016994 Simplifying x = 1.809016994Subproblem 2
x + -1.25 = -0.559016994 Simplifying x + -1.25 = -0.559016994 Reorder the terms: -1.25 + x = -0.559016994 Solving -1.25 + x = -0.559016994 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -0.559016994 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -0.559016994 + 1.25 x = -0.559016994 + 1.25 Combine like terms: -0.559016994 + 1.25 = 0.690983006 x = 0.690983006 Simplifying x = 0.690983006Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.809016994, 0.690983006}
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