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Simplifying 16(9x + -1)(1 + -1x) + -1(4 + -1x)(74x + -8) = 48 Reorder the terms: 16(-1 + 9x)(1 + -1x) + -1(4 + -1x)(74x + -8) = 48 Multiply (-1 + 9x) * (1 + -1x) 16(-1(1 + -1x) + 9x * (1 + -1x)) + -1(4 + -1x)(74x + -8) = 48 16((1 * -1 + -1x * -1) + 9x * (1 + -1x)) + -1(4 + -1x)(74x + -8) = 48 16((-1 + 1x) + 9x * (1 + -1x)) + -1(4 + -1x)(74x + -8) = 48 16(-1 + 1x + (1 * 9x + -1x * 9x)) + -1(4 + -1x)(74x + -8) = 48 16(-1 + 1x + (9x + -9x2)) + -1(4 + -1x)(74x + -8) = 48 Combine like terms: 1x + 9x = 10x 16(-1 + 10x + -9x2) + -1(4 + -1x)(74x + -8) = 48 (-1 * 16 + 10x * 16 + -9x2 * 16) + -1(4 + -1x)(74x + -8) = 48 (-16 + 160x + -144x2) + -1(4 + -1x)(74x + -8) = 48 Reorder the terms: -16 + 160x + -144x2 + -1(4 + -1x)(-8 + 74x) = 48 Multiply (4 + -1x) * (-8 + 74x) -16 + 160x + -144x2 + -1(4(-8 + 74x) + -1x * (-8 + 74x)) = 48 -16 + 160x + -144x2 + -1((-8 * 4 + 74x * 4) + -1x * (-8 + 74x)) = 48 -16 + 160x + -144x2 + -1((-32 + 296x) + -1x * (-8 + 74x)) = 48 -16 + 160x + -144x2 + -1(-32 + 296x + (-8 * -1x + 74x * -1x)) = 48 -16 + 160x + -144x2 + -1(-32 + 296x + (8x + -74x2)) = 48 Combine like terms: 296x + 8x = 304x -16 + 160x + -144x2 + -1(-32 + 304x + -74x2) = 48 -16 + 160x + -144x2 + (-32 * -1 + 304x * -1 + -74x2 * -1) = 48 -16 + 160x + -144x2 + (32 + -304x + 74x2) = 48 Reorder the terms: -16 + 32 + 160x + -304x + -144x2 + 74x2 = 48 Combine like terms: -16 + 32 = 16 16 + 160x + -304x + -144x2 + 74x2 = 48 Combine like terms: 160x + -304x = -144x 16 + -144x + -144x2 + 74x2 = 48 Combine like terms: -144x2 + 74x2 = -70x2 16 + -144x + -70x2 = 48 Solving 16 + -144x + -70x2 = 48 Solving for variable 'x'. Reorder the terms: 16 + -48 + -144x + -70x2 = 48 + -48 Combine like terms: 16 + -48 = -32 -32 + -144x + -70x2 = 48 + -48 Combine like terms: 48 + -48 = 0 -32 + -144x + -70x2 = 0 Factor out the Greatest Common Factor (GCF), '-2'. -2(16 + 72x + 35x2) = 0 Ignore the factor -2.Subproblem 1
Set the factor '(16 + 72x + 35x2)' equal to zero and attempt to solve: Simplifying 16 + 72x + 35x2 = 0 Solving 16 + 72x + 35x2 = 0 Begin completing the square. Divide all terms by 35 the coefficient of the squared term: Divide each side by '35'. 0.4571428571 + 2.057142857x + x2 = 0 Move the constant term to the right: Add '-0.4571428571' to each side of the equation. 0.4571428571 + 2.057142857x + -0.4571428571 + x2 = 0 + -0.4571428571 Reorder the terms: 0.4571428571 + -0.4571428571 + 2.057142857x + x2 = 0 + -0.4571428571 Combine like terms: 0.4571428571 + -0.4571428571 = 0.0000000000 0.0000000000 + 2.057142857x + x2 = 0 + -0.4571428571 2.057142857x + x2 = 0 + -0.4571428571 Combine like terms: 0 + -0.4571428571 = -0.4571428571 2.057142857x + x2 = -0.4571428571 The x term is 2.057142857x. Take half its coefficient (1.028571429). Square it (1.057959185) and add it to both sides. Add '1.057959185' to each side of the equation. 2.057142857x + 1.057959185 + x2 = -0.4571428571 + 1.057959185 Reorder the terms: 1.057959185 + 2.057142857x + x2 = -0.4571428571 + 1.057959185 Combine like terms: -0.4571428571 + 1.057959185 = 0.6008163279 1.057959185 + 2.057142857x + x2 = 0.6008163279 Factor a perfect square on the left side: (x + 1.028571429)(x + 1.028571429) = 0.6008163279 Calculate the square root of the right side: 0.775123428 Break this problem into two subproblems by setting (x + 1.028571429) equal to 0.775123428 and -0.775123428.Subproblem 1
x + 1.028571429 = 0.775123428 Simplifying x + 1.028571429 = 0.775123428 Reorder the terms: 1.028571429 + x = 0.775123428 Solving 1.028571429 + x = 0.775123428 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.028571429' to each side of the equation. 1.028571429 + -1.028571429 + x = 0.775123428 + -1.028571429 Combine like terms: 1.028571429 + -1.028571429 = 0.000000000 0.000000000 + x = 0.775123428 + -1.028571429 x = 0.775123428 + -1.028571429 Combine like terms: 0.775123428 + -1.028571429 = -0.253448001 x = -0.253448001 Simplifying x = -0.253448001Subproblem 2
x + 1.028571429 = -0.775123428 Simplifying x + 1.028571429 = -0.775123428 Reorder the terms: 1.028571429 + x = -0.775123428 Solving 1.028571429 + x = -0.775123428 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.028571429' to each side of the equation. 1.028571429 + -1.028571429 + x = -0.775123428 + -1.028571429 Combine like terms: 1.028571429 + -1.028571429 = 0.000000000 0.000000000 + x = -0.775123428 + -1.028571429 x = -0.775123428 + -1.028571429 Combine like terms: -0.775123428 + -1.028571429 = -1.803694857 x = -1.803694857 Simplifying x = -1.803694857Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.253448001, -1.803694857}Solution
x = {-0.253448001, -1.803694857}
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