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Simplifying 5x(x + -1) + 5 = 7 + x(1 + -2x) Reorder the terms: 5x(-1 + x) + 5 = 7 + x(1 + -2x) (-1 * 5x + x * 5x) + 5 = 7 + x(1 + -2x) (-5x + 5x2) + 5 = 7 + x(1 + -2x) Reorder the terms: 5 + -5x + 5x2 = 7 + x(1 + -2x) 5 + -5x + 5x2 = 7 + (1 * x + -2x * x) 5 + -5x + 5x2 = 7 + (1x + -2x2) Solving 5 + -5x + 5x2 = 7 + 1x + -2x2 Solving for variable 'x'. Reorder the terms: 5 + -7 + -5x + -1x + 5x2 + 2x2 = 7 + 1x + -2x2 + -7 + -1x + 2x2 Combine like terms: 5 + -7 = -2 -2 + -5x + -1x + 5x2 + 2x2 = 7 + 1x + -2x2 + -7 + -1x + 2x2 Combine like terms: -5x + -1x = -6x -2 + -6x + 5x2 + 2x2 = 7 + 1x + -2x2 + -7 + -1x + 2x2 Combine like terms: 5x2 + 2x2 = 7x2 -2 + -6x + 7x2 = 7 + 1x + -2x2 + -7 + -1x + 2x2 Reorder the terms: -2 + -6x + 7x2 = 7 + -7 + 1x + -1x + -2x2 + 2x2 Combine like terms: 7 + -7 = 0 -2 + -6x + 7x2 = 0 + 1x + -1x + -2x2 + 2x2 -2 + -6x + 7x2 = 1x + -1x + -2x2 + 2x2 Combine like terms: 1x + -1x = 0 -2 + -6x + 7x2 = 0 + -2x2 + 2x2 -2 + -6x + 7x2 = -2x2 + 2x2 Combine like terms: -2x2 + 2x2 = 0 -2 + -6x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.2857142857 + -0.8571428571x + x2 = 0 Move the constant term to the right: Add '0.2857142857' to each side of the equation. -0.2857142857 + -0.8571428571x + 0.2857142857 + x2 = 0 + 0.2857142857 Reorder the terms: -0.2857142857 + 0.2857142857 + -0.8571428571x + x2 = 0 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + -0.8571428571x + x2 = 0 + 0.2857142857 -0.8571428571x + x2 = 0 + 0.2857142857 Combine like terms: 0 + 0.2857142857 = 0.2857142857 -0.8571428571x + x2 = 0.2857142857 The x term is -0.8571428571x. Take half its coefficient (-0.4285714286). Square it (0.1836734694) and add it to both sides. Add '0.1836734694' to each side of the equation. -0.8571428571x + 0.1836734694 + x2 = 0.2857142857 + 0.1836734694 Reorder the terms: 0.1836734694 + -0.8571428571x + x2 = 0.2857142857 + 0.1836734694 Combine like terms: 0.2857142857 + 0.1836734694 = 0.4693877551 0.1836734694 + -0.8571428571x + x2 = 0.4693877551 Factor a perfect square on the left side: (x + -0.4285714286)(x + -0.4285714286) = 0.4693877551 Calculate the square root of the right side: 0.685118789 Break this problem into two subproblems by setting (x + -0.4285714286) equal to 0.685118789 and -0.685118789.Subproblem 1
x + -0.4285714286 = 0.685118789 Simplifying x + -0.4285714286 = 0.685118789 Reorder the terms: -0.4285714286 + x = 0.685118789 Solving -0.4285714286 + x = 0.685118789 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + x = 0.685118789 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + x = 0.685118789 + 0.4285714286 x = 0.685118789 + 0.4285714286 Combine like terms: 0.685118789 + 0.4285714286 = 1.1136902176 x = 1.1136902176 Simplifying x = 1.1136902176Subproblem 2
x + -0.4285714286 = -0.685118789 Simplifying x + -0.4285714286 = -0.685118789 Reorder the terms: -0.4285714286 + x = -0.685118789 Solving -0.4285714286 + x = -0.685118789 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4285714286' to each side of the equation. -0.4285714286 + 0.4285714286 + x = -0.685118789 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + x = -0.685118789 + 0.4285714286 x = -0.685118789 + 0.4285714286 Combine like terms: -0.685118789 + 0.4285714286 = -0.2565473604 x = -0.2565473604 Simplifying x = -0.2565473604Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.1136902176, -0.2565473604}
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