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Simplifying 100 = (2x + 5)(2x + 10) Reorder the terms: 100 = (5 + 2x)(2x + 10) Reorder the terms: 100 = (5 + 2x)(10 + 2x) Multiply (5 + 2x) * (10 + 2x) 100 = (5(10 + 2x) + 2x * (10 + 2x)) 100 = ((10 * 5 + 2x * 5) + 2x * (10 + 2x)) 100 = ((50 + 10x) + 2x * (10 + 2x)) 100 = (50 + 10x + (10 * 2x + 2x * 2x)) 100 = (50 + 10x + (20x + 4x2)) Combine like terms: 10x + 20x = 30x 100 = (50 + 30x + 4x2) Solving 100 = 50 + 30x + 4x2 Solving for variable 'x'. Combine like terms: 100 + -50 = 50 50 + -30x + -4x2 = 50 + 30x + 4x2 + -50 + -30x + -4x2 Reorder the terms: 50 + -30x + -4x2 = 50 + -50 + 30x + -30x + 4x2 + -4x2 Combine like terms: 50 + -50 = 0 50 + -30x + -4x2 = 0 + 30x + -30x + 4x2 + -4x2 50 + -30x + -4x2 = 30x + -30x + 4x2 + -4x2 Combine like terms: 30x + -30x = 0 50 + -30x + -4x2 = 0 + 4x2 + -4x2 50 + -30x + -4x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 50 + -30x + -4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(25 + -15x + -2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(25 + -15x + -2x2)' equal to zero and attempt to solve: Simplifying 25 + -15x + -2x2 = 0 Solving 25 + -15x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -12.5 + 7.5x + x2 = 0 Move the constant term to the right: Add '12.5' to each side of the equation. -12.5 + 7.5x + 12.5 + x2 = 0 + 12.5 Reorder the terms: -12.5 + 12.5 + 7.5x + x2 = 0 + 12.5 Combine like terms: -12.5 + 12.5 = 0.0 0.0 + 7.5x + x2 = 0 + 12.5 7.5x + x2 = 0 + 12.5 Combine like terms: 0 + 12.5 = 12.5 7.5x + x2 = 12.5 The x term is 7.5x. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5x + 14.0625 + x2 = 12.5 + 14.0625 Reorder the terms: 14.0625 + 7.5x + x2 = 12.5 + 14.0625 Combine like terms: 12.5 + 14.0625 = 26.5625 14.0625 + 7.5x + x2 = 26.5625 Factor a perfect square on the left side: (x + 3.75)(x + 3.75) = 26.5625 Calculate the square root of the right side: 5.153882032 Break this problem into two subproblems by setting (x + 3.75) equal to 5.153882032 and -5.153882032.Subproblem 1
x + 3.75 = 5.153882032 Simplifying x + 3.75 = 5.153882032 Reorder the terms: 3.75 + x = 5.153882032 Solving 3.75 + x = 5.153882032 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + x = 5.153882032 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + x = 5.153882032 + -3.75 x = 5.153882032 + -3.75 Combine like terms: 5.153882032 + -3.75 = 1.403882032 x = 1.403882032 Simplifying x = 1.403882032Subproblem 2
x + 3.75 = -5.153882032 Simplifying x + 3.75 = -5.153882032 Reorder the terms: 3.75 + x = -5.153882032 Solving 3.75 + x = -5.153882032 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + x = -5.153882032 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + x = -5.153882032 + -3.75 x = -5.153882032 + -3.75 Combine like terms: -5.153882032 + -3.75 = -8.903882032 x = -8.903882032 Simplifying x = -8.903882032Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.403882032, -8.903882032}Solution
x = {1.403882032, -8.903882032}
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