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Simplifying 4x(x + 1) = 75 + x(x + 7) Reorder the terms: 4x(1 + x) = 75 + x(x + 7) (1 * 4x + x * 4x) = 75 + x(x + 7) (4x + 4x2) = 75 + x(x + 7) Reorder the terms: 4x + 4x2 = 75 + x(7 + x) 4x + 4x2 = 75 + (7 * x + x * x) 4x + 4x2 = 75 + (7x + x2) Solving 4x + 4x2 = 75 + 7x + x2 Solving for variable 'x'. Reorder the terms: -75 + 4x + -7x + 4x2 + -1x2 = 75 + 7x + x2 + -75 + -7x + -1x2 Combine like terms: 4x + -7x = -3x -75 + -3x + 4x2 + -1x2 = 75 + 7x + x2 + -75 + -7x + -1x2 Combine like terms: 4x2 + -1x2 = 3x2 -75 + -3x + 3x2 = 75 + 7x + x2 + -75 + -7x + -1x2 Reorder the terms: -75 + -3x + 3x2 = 75 + -75 + 7x + -7x + x2 + -1x2 Combine like terms: 75 + -75 = 0 -75 + -3x + 3x2 = 0 + 7x + -7x + x2 + -1x2 -75 + -3x + 3x2 = 7x + -7x + x2 + -1x2 Combine like terms: 7x + -7x = 0 -75 + -3x + 3x2 = 0 + x2 + -1x2 -75 + -3x + 3x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -75 + -3x + 3x2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-25 + -1x + x2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-25 + -1x + x2)' equal to zero and attempt to solve: Simplifying -25 + -1x + x2 = 0 Solving -25 + -1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '25' to each side of the equation. -25 + -1x + 25 + x2 = 0 + 25 Reorder the terms: -25 + 25 + -1x + x2 = 0 + 25 Combine like terms: -25 + 25 = 0 0 + -1x + x2 = 0 + 25 -1x + x2 = 0 + 25 Combine like terms: 0 + 25 = 25 -1x + x2 = 25 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = 25 + 0.25 Reorder the terms: 0.25 + -1x + x2 = 25 + 0.25 Combine like terms: 25 + 0.25 = 25.25 0.25 + -1x + x2 = 25.25 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 25.25 Calculate the square root of the right side: 5.024937811 Break this problem into two subproblems by setting (x + -0.5) equal to 5.024937811 and -5.024937811.Subproblem 1
x + -0.5 = 5.024937811 Simplifying x + -0.5 = 5.024937811 Reorder the terms: -0.5 + x = 5.024937811 Solving -0.5 + x = 5.024937811 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 5.024937811 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 5.024937811 + 0.5 x = 5.024937811 + 0.5 Combine like terms: 5.024937811 + 0.5 = 5.524937811 x = 5.524937811 Simplifying x = 5.524937811Subproblem 2
x + -0.5 = -5.024937811 Simplifying x + -0.5 = -5.024937811 Reorder the terms: -0.5 + x = -5.024937811 Solving -0.5 + x = -5.024937811 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -5.024937811 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -5.024937811 + 0.5 x = -5.024937811 + 0.5 Combine like terms: -5.024937811 + 0.5 = -4.524937811 x = -4.524937811 Simplifying x = -4.524937811Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.524937811, -4.524937811}Solution
x = {5.524937811, -4.524937811}
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