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Simplifying (3x + 4)(11x + -6) = 1 Reorder the terms: (4 + 3x)(11x + -6) = 1 Reorder the terms: (4 + 3x)(-6 + 11x) = 1 Multiply (4 + 3x) * (-6 + 11x) (4(-6 + 11x) + 3x * (-6 + 11x)) = 1 ((-6 * 4 + 11x * 4) + 3x * (-6 + 11x)) = 1 ((-24 + 44x) + 3x * (-6 + 11x)) = 1 (-24 + 44x + (-6 * 3x + 11x * 3x)) = 1 (-24 + 44x + (-18x + 33x2)) = 1 Combine like terms: 44x + -18x = 26x (-24 + 26x + 33x2) = 1 Solving -24 + 26x + 33x2 = 1 Solving for variable 'x'. Reorder the terms: -24 + -1 + 26x + 33x2 = 1 + -1 Combine like terms: -24 + -1 = -25 -25 + 26x + 33x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -25 + 26x + 33x2 = 0 Begin completing the square. Divide all terms by 33 the coefficient of the squared term: Divide each side by '33'. -0.7575757576 + 0.7878787879x + x2 = 0 Move the constant term to the right: Add '0.7575757576' to each side of the equation. -0.7575757576 + 0.7878787879x + 0.7575757576 + x2 = 0 + 0.7575757576 Reorder the terms: -0.7575757576 + 0.7575757576 + 0.7878787879x + x2 = 0 + 0.7575757576 Combine like terms: -0.7575757576 + 0.7575757576 = 0.0000000000 0.0000000000 + 0.7878787879x + x2 = 0 + 0.7575757576 0.7878787879x + x2 = 0 + 0.7575757576 Combine like terms: 0 + 0.7575757576 = 0.7575757576 0.7878787879x + x2 = 0.7575757576 The x term is 0.7878787879x. Take half its coefficient (0.393939394). Square it (0.1551882461) and add it to both sides. Add '0.1551882461' to each side of the equation. 0.7878787879x + 0.1551882461 + x2 = 0.7575757576 + 0.1551882461 Reorder the terms: 0.1551882461 + 0.7878787879x + x2 = 0.7575757576 + 0.1551882461 Combine like terms: 0.7575757576 + 0.1551882461 = 0.9127640037 0.1551882461 + 0.7878787879x + x2 = 0.9127640037 Factor a perfect square on the left side: (x + 0.393939394)(x + 0.393939394) = 0.9127640037 Calculate the square root of the right side: 0.955386835 Break this problem into two subproblems by setting (x + 0.393939394) equal to 0.955386835 and -0.955386835.Subproblem 1
x + 0.393939394 = 0.955386835 Simplifying x + 0.393939394 = 0.955386835 Reorder the terms: 0.393939394 + x = 0.955386835 Solving 0.393939394 + x = 0.955386835 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.393939394' to each side of the equation. 0.393939394 + -0.393939394 + x = 0.955386835 + -0.393939394 Combine like terms: 0.393939394 + -0.393939394 = 0.000000000 0.000000000 + x = 0.955386835 + -0.393939394 x = 0.955386835 + -0.393939394 Combine like terms: 0.955386835 + -0.393939394 = 0.561447441 x = 0.561447441 Simplifying x = 0.561447441Subproblem 2
x + 0.393939394 = -0.955386835 Simplifying x + 0.393939394 = -0.955386835 Reorder the terms: 0.393939394 + x = -0.955386835 Solving 0.393939394 + x = -0.955386835 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.393939394' to each side of the equation. 0.393939394 + -0.393939394 + x = -0.955386835 + -0.393939394 Combine like terms: 0.393939394 + -0.393939394 = 0.000000000 0.000000000 + x = -0.955386835 + -0.393939394 x = -0.955386835 + -0.393939394 Combine like terms: -0.955386835 + -0.393939394 = -1.349326229 x = -1.349326229 Simplifying x = -1.349326229Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.561447441, -1.349326229}
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