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Simplifying 75 = (10 + 2x)(5 + 2x) Multiply (10 + 2x) * (5 + 2x) 75 = (10(5 + 2x) + 2x * (5 + 2x)) 75 = ((5 * 10 + 2x * 10) + 2x * (5 + 2x)) 75 = ((50 + 20x) + 2x * (5 + 2x)) 75 = (50 + 20x + (5 * 2x + 2x * 2x)) 75 = (50 + 20x + (10x + 4x2)) Combine like terms: 20x + 10x = 30x 75 = (50 + 30x + 4x2) Solving 75 = 50 + 30x + 4x2 Solving for variable 'x'. Combine like terms: 75 + -50 = 25 25 + -30x + -4x2 = 50 + 30x + 4x2 + -50 + -30x + -4x2 Reorder the terms: 25 + -30x + -4x2 = 50 + -50 + 30x + -30x + 4x2 + -4x2 Combine like terms: 50 + -50 = 0 25 + -30x + -4x2 = 0 + 30x + -30x + 4x2 + -4x2 25 + -30x + -4x2 = 30x + -30x + 4x2 + -4x2 Combine like terms: 30x + -30x = 0 25 + -30x + -4x2 = 0 + 4x2 + -4x2 25 + -30x + -4x2 = 4x2 + -4x2 Combine like terms: 4x2 + -4x2 = 0 25 + -30x + -4x2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -6.25 + 7.5x + x2 = 0 Move the constant term to the right: Add '6.25' to each side of the equation. -6.25 + 7.5x + 6.25 + x2 = 0 + 6.25 Reorder the terms: -6.25 + 6.25 + 7.5x + x2 = 0 + 6.25 Combine like terms: -6.25 + 6.25 = 0.00 0.00 + 7.5x + x2 = 0 + 6.25 7.5x + x2 = 0 + 6.25 Combine like terms: 0 + 6.25 = 6.25 7.5x + x2 = 6.25 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 = 6.25 + 14.0625 Reorder the terms: 14.0625 + 7.5x + x2 = 6.25 + 14.0625 Combine like terms: 6.25 + 14.0625 = 20.3125 14.0625 + 7.5x + x2 = 20.3125 Factor a perfect square on the left side: (x + 3.75)(x + 3.75) = 20.3125 Calculate the square root of the right side: 4.506939094 Break this problem into two subproblems by setting (x + 3.75) equal to 4.506939094 and -4.506939094.Subproblem 1
x + 3.75 = 4.506939094 Simplifying x + 3.75 = 4.506939094 Reorder the terms: 3.75 + x = 4.506939094 Solving 3.75 + x = 4.506939094 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 = 4.506939094 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + x = 4.506939094 + -3.75 x = 4.506939094 + -3.75 Combine like terms: 4.506939094 + -3.75 = 0.756939094 x = 0.756939094 Simplifying x = 0.756939094Subproblem 2
x + 3.75 = -4.506939094 Simplifying x + 3.75 = -4.506939094 Reorder the terms: 3.75 + x = -4.506939094 Solving 3.75 + x = -4.506939094 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 = -4.506939094 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + x = -4.506939094 + -3.75 x = -4.506939094 + -3.75 Combine like terms: -4.506939094 + -3.75 = -8.256939094 x = -8.256939094 Simplifying x = -8.256939094Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.756939094, -8.256939094}
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