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Simplifying (61x + -84)(43x + -63) = 869110 Reorder the terms: (-84 + 61x)(43x + -63) = 869110 Reorder the terms: (-84 + 61x)(-63 + 43x) = 869110 Multiply (-84 + 61x) * (-63 + 43x) (-84(-63 + 43x) + 61x * (-63 + 43x)) = 869110 ((-63 * -84 + 43x * -84) + 61x * (-63 + 43x)) = 869110 ((5292 + -3612x) + 61x * (-63 + 43x)) = 869110 (5292 + -3612x + (-63 * 61x + 43x * 61x)) = 869110 (5292 + -3612x + (-3843x + 2623x2)) = 869110 Combine like terms: -3612x + -3843x = -7455x (5292 + -7455x + 2623x2) = 869110 Solving 5292 + -7455x + 2623x2 = 869110 Solving for variable 'x'. Reorder the terms: 5292 + -869110 + -7455x + 2623x2 = 869110 + -869110 Combine like terms: 5292 + -869110 = -863818 -863818 + -7455x + 2623x2 = 869110 + -869110 Combine like terms: 869110 + -869110 = 0 -863818 + -7455x + 2623x2 = 0 Begin completing the square. Divide all terms by 2623 the coefficient of the squared term: Divide each side by '2623'. -329.3244377 + -2.842165459x + x2 = 0 Move the constant term to the right: Add '329.3244377' to each side of the equation. -329.3244377 + -2.842165459x + 329.3244377 + x2 = 0 + 329.3244377 Reorder the terms: -329.3244377 + 329.3244377 + -2.842165459x + x2 = 0 + 329.3244377 Combine like terms: -329.3244377 + 329.3244377 = 0.0000000 0.0000000 + -2.842165459x + x2 = 0 + 329.3244377 -2.842165459x + x2 = 0 + 329.3244377 Combine like terms: 0 + 329.3244377 = 329.3244377 -2.842165459x + x2 = 329.3244377 The x term is -2.842165459x. Take half its coefficient (-1.42108273). Square it (2.019476126) and add it to both sides. Add '2.019476126' to each side of the equation. -2.842165459x + 2.019476126 + x2 = 329.3244377 + 2.019476126 Reorder the terms: 2.019476126 + -2.842165459x + x2 = 329.3244377 + 2.019476126 Combine like terms: 329.3244377 + 2.019476126 = 331.343913826 2.019476126 + -2.842165459x + x2 = 331.343913826 Factor a perfect square on the left side: (x + -1.42108273)(x + -1.42108273) = 331.343913826 Calculate the square root of the right side: 18.202854552 Break this problem into two subproblems by setting (x + -1.42108273) equal to 18.202854552 and -18.202854552.Subproblem 1
x + -1.42108273 = 18.202854552 Simplifying x + -1.42108273 = 18.202854552 Reorder the terms: -1.42108273 + x = 18.202854552 Solving -1.42108273 + x = 18.202854552 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.42108273' to each side of the equation. -1.42108273 + 1.42108273 + x = 18.202854552 + 1.42108273 Combine like terms: -1.42108273 + 1.42108273 = 0.00000000 0.00000000 + x = 18.202854552 + 1.42108273 x = 18.202854552 + 1.42108273 Combine like terms: 18.202854552 + 1.42108273 = 19.623937282 x = 19.623937282 Simplifying x = 19.623937282Subproblem 2
x + -1.42108273 = -18.202854552 Simplifying x + -1.42108273 = -18.202854552 Reorder the terms: -1.42108273 + x = -18.202854552 Solving -1.42108273 + x = -18.202854552 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.42108273' to each side of the equation. -1.42108273 + 1.42108273 + x = -18.202854552 + 1.42108273 Combine like terms: -1.42108273 + 1.42108273 = 0.00000000 0.00000000 + x = -18.202854552 + 1.42108273 x = -18.202854552 + 1.42108273 Combine like terms: -18.202854552 + 1.42108273 = -16.781771822 x = -16.781771822 Simplifying x = -16.781771822Solution
The solution to the problem is based on the solutions from the subproblems. x = {19.623937282, -16.781771822}
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