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Simplifying 315 = (x + 2)(x + 5) Reorder the terms: 315 = (2 + x)(x + 5) Reorder the terms: 315 = (2 + x)(5 + x) Multiply (2 + x) * (5 + x) 315 = (2(5 + x) + x(5 + x)) 315 = ((5 * 2 + x * 2) + x(5 + x)) 315 = ((10 + 2x) + x(5 + x)) 315 = (10 + 2x + (5 * x + x * x)) 315 = (10 + 2x + (5x + x2)) Combine like terms: 2x + 5x = 7x 315 = (10 + 7x + x2) Solving 315 = 10 + 7x + x2 Solving for variable 'x'. Combine like terms: 315 + -10 = 305 305 + -7x + -1x2 = 10 + 7x + x2 + -10 + -7x + -1x2 Reorder the terms: 305 + -7x + -1x2 = 10 + -10 + 7x + -7x + x2 + -1x2 Combine like terms: 10 + -10 = 0 305 + -7x + -1x2 = 0 + 7x + -7x + x2 + -1x2 305 + -7x + -1x2 = 7x + -7x + x2 + -1x2 Combine like terms: 7x + -7x = 0 305 + -7x + -1x2 = 0 + x2 + -1x2 305 + -7x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 305 + -7x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -305 + 7x + x2 = 0 Move the constant term to the right: Add '305' to each side of the equation. -305 + 7x + 305 + x2 = 0 + 305 Reorder the terms: -305 + 305 + 7x + x2 = 0 + 305 Combine like terms: -305 + 305 = 0 0 + 7x + x2 = 0 + 305 7x + x2 = 0 + 305 Combine like terms: 0 + 305 = 305 7x + x2 = 305 The x term is 7x. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7x + 12.25 + x2 = 305 + 12.25 Reorder the terms: 12.25 + 7x + x2 = 305 + 12.25 Combine like terms: 305 + 12.25 = 317.25 12.25 + 7x + x2 = 317.25 Factor a perfect square on the left side: (x + 3.5)(x + 3.5) = 317.25 Calculate the square root of the right side: 17.811513131 Break this problem into two subproblems by setting (x + 3.5) equal to 17.811513131 and -17.811513131.Subproblem 1
x + 3.5 = 17.811513131 Simplifying x + 3.5 = 17.811513131 Reorder the terms: 3.5 + x = 17.811513131 Solving 3.5 + x = 17.811513131 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 17.811513131 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 17.811513131 + -3.5 x = 17.811513131 + -3.5 Combine like terms: 17.811513131 + -3.5 = 14.311513131 x = 14.311513131 Simplifying x = 14.311513131Subproblem 2
x + 3.5 = -17.811513131 Simplifying x + 3.5 = -17.811513131 Reorder the terms: 3.5 + x = -17.811513131 Solving 3.5 + x = -17.811513131 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -17.811513131 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -17.811513131 + -3.5 x = -17.811513131 + -3.5 Combine like terms: -17.811513131 + -3.5 = -21.311513131 x = -21.311513131 Simplifying x = -21.311513131Solution
The solution to the problem is based on the solutions from the subproblems. x = {14.311513131, -21.311513131}
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