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Simplifying 210 = (2x + 7)(x + 1) Reorder the terms: 210 = (7 + 2x)(x + 1) Reorder the terms: 210 = (7 + 2x)(1 + x) Multiply (7 + 2x) * (1 + x) 210 = (7(1 + x) + 2x * (1 + x)) 210 = ((1 * 7 + x * 7) + 2x * (1 + x)) 210 = ((7 + 7x) + 2x * (1 + x)) 210 = (7 + 7x + (1 * 2x + x * 2x)) 210 = (7 + 7x + (2x + 2x2)) Combine like terms: 7x + 2x = 9x 210 = (7 + 9x + 2x2) Solving 210 = 7 + 9x + 2x2 Solving for variable 'x'. Combine like terms: 210 + -7 = 203 203 + -9x + -2x2 = 7 + 9x + 2x2 + -7 + -9x + -2x2 Reorder the terms: 203 + -9x + -2x2 = 7 + -7 + 9x + -9x + 2x2 + -2x2 Combine like terms: 7 + -7 = 0 203 + -9x + -2x2 = 0 + 9x + -9x + 2x2 + -2x2 203 + -9x + -2x2 = 9x + -9x + 2x2 + -2x2 Combine like terms: 9x + -9x = 0 203 + -9x + -2x2 = 0 + 2x2 + -2x2 203 + -9x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 203 + -9x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -101.5 + 4.5x + x2 = 0 Move the constant term to the right: Add '101.5' to each side of the equation. -101.5 + 4.5x + 101.5 + x2 = 0 + 101.5 Reorder the terms: -101.5 + 101.5 + 4.5x + x2 = 0 + 101.5 Combine like terms: -101.5 + 101.5 = 0.0 0.0 + 4.5x + x2 = 0 + 101.5 4.5x + x2 = 0 + 101.5 Combine like terms: 0 + 101.5 = 101.5 4.5x + x2 = 101.5 The x term is 4.5x. Take half its coefficient (2.25). Square it (5.0625) and add it to both sides. Add '5.0625' to each side of the equation. 4.5x + 5.0625 + x2 = 101.5 + 5.0625 Reorder the terms: 5.0625 + 4.5x + x2 = 101.5 + 5.0625 Combine like terms: 101.5 + 5.0625 = 106.5625 5.0625 + 4.5x + x2 = 106.5625 Factor a perfect square on the left side: (x + 2.25)(x + 2.25) = 106.5625 Calculate the square root of the right side: 10.322911411 Break this problem into two subproblems by setting (x + 2.25) equal to 10.322911411 and -10.322911411.Subproblem 1
x + 2.25 = 10.322911411 Simplifying x + 2.25 = 10.322911411 Reorder the terms: 2.25 + x = 10.322911411 Solving 2.25 + x = 10.322911411 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = 10.322911411 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = 10.322911411 + -2.25 x = 10.322911411 + -2.25 Combine like terms: 10.322911411 + -2.25 = 8.072911411 x = 8.072911411 Simplifying x = 8.072911411Subproblem 2
x + 2.25 = -10.322911411 Simplifying x + 2.25 = -10.322911411 Reorder the terms: 2.25 + x = -10.322911411 Solving 2.25 + x = -10.322911411 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.25' to each side of the equation. 2.25 + -2.25 + x = -10.322911411 + -2.25 Combine like terms: 2.25 + -2.25 = 0.00 0.00 + x = -10.322911411 + -2.25 x = -10.322911411 + -2.25 Combine like terms: -10.322911411 + -2.25 = -12.572911411 x = -12.572911411 Simplifying x = -12.572911411Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.072911411, -12.572911411}
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