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Simplifying x(2x + 5) = 4x + 7 Reorder the terms: x(5 + 2x) = 4x + 7 (5 * x + 2x * x) = 4x + 7 (5x + 2x2) = 4x + 7 Reorder the terms: 5x + 2x2 = 7 + 4x Solving 5x + 2x2 = 7 + 4x Solving for variable 'x'. Reorder the terms: -7 + 5x + -4x + 2x2 = 7 + 4x + -7 + -4x Combine like terms: 5x + -4x = 1x -7 + 1x + 2x2 = 7 + 4x + -7 + -4x Reorder the terms: -7 + 1x + 2x2 = 7 + -7 + 4x + -4x Combine like terms: 7 + -7 = 0 -7 + 1x + 2x2 = 0 + 4x + -4x -7 + 1x + 2x2 = 4x + -4x Combine like terms: 4x + -4x = 0 -7 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + 0.5x + x2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 0.5x + 3.5 + x2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 0.5x + x2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 0.5x + x2 = 0 + 3.5 0.5x + x2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 0.5x + x2 = 3.5 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 3.5 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 3.5 + 0.0625 Combine like terms: 3.5 + 0.0625 = 3.5625 0.0625 + 0.5x + x2 = 3.5625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 3.5625 Calculate the square root of the right side: 1.887458609 Break this problem into two subproblems by setting (x + 0.25) equal to 1.887458609 and -1.887458609.Subproblem 1
x + 0.25 = 1.887458609 Simplifying x + 0.25 = 1.887458609 Reorder the terms: 0.25 + x = 1.887458609 Solving 0.25 + x = 1.887458609 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 1.887458609 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 1.887458609 + -0.25 x = 1.887458609 + -0.25 Combine like terms: 1.887458609 + -0.25 = 1.637458609 x = 1.637458609 Simplifying x = 1.637458609Subproblem 2
x + 0.25 = -1.887458609 Simplifying x + 0.25 = -1.887458609 Reorder the terms: 0.25 + x = -1.887458609 Solving 0.25 + x = -1.887458609 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -1.887458609 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -1.887458609 + -0.25 x = -1.887458609 + -0.25 Combine like terms: -1.887458609 + -0.25 = -2.137458609 x = -2.137458609 Simplifying x = -2.137458609Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.637458609, -2.137458609}
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