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Simplifying (4x + 2)(-12x + 8) + 2(5x + -3) = 0 Reorder the terms: (2 + 4x)(-12x + 8) + 2(5x + -3) = 0 Reorder the terms: (2 + 4x)(8 + -12x) + 2(5x + -3) = 0 Multiply (2 + 4x) * (8 + -12x) (2(8 + -12x) + 4x * (8 + -12x)) + 2(5x + -3) = 0 ((8 * 2 + -12x * 2) + 4x * (8 + -12x)) + 2(5x + -3) = 0 ((16 + -24x) + 4x * (8 + -12x)) + 2(5x + -3) = 0 (16 + -24x + (8 * 4x + -12x * 4x)) + 2(5x + -3) = 0 (16 + -24x + (32x + -48x2)) + 2(5x + -3) = 0 Combine like terms: -24x + 32x = 8x (16 + 8x + -48x2) + 2(5x + -3) = 0 Reorder the terms: 16 + 8x + -48x2 + 2(-3 + 5x) = 0 16 + 8x + -48x2 + (-3 * 2 + 5x * 2) = 0 16 + 8x + -48x2 + (-6 + 10x) = 0 Reorder the terms: 16 + -6 + 8x + 10x + -48x2 = 0 Combine like terms: 16 + -6 = 10 10 + 8x + 10x + -48x2 = 0 Combine like terms: 8x + 10x = 18x 10 + 18x + -48x2 = 0 Solving 10 + 18x + -48x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(5 + 9x + -24x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(5 + 9x + -24x2)' equal to zero and attempt to solve: Simplifying 5 + 9x + -24x2 = 0 Solving 5 + 9x + -24x2 = 0 Begin completing the square. Divide all terms by -24 the coefficient of the squared term: Divide each side by '-24'. -0.2083333333 + -0.375x + x2 = 0 Move the constant term to the right: Add '0.2083333333' to each side of the equation. -0.2083333333 + -0.375x + 0.2083333333 + x2 = 0 + 0.2083333333 Reorder the terms: -0.2083333333 + 0.2083333333 + -0.375x + x2 = 0 + 0.2083333333 Combine like terms: -0.2083333333 + 0.2083333333 = 0.0000000000 0.0000000000 + -0.375x + x2 = 0 + 0.2083333333 -0.375x + x2 = 0 + 0.2083333333 Combine like terms: 0 + 0.2083333333 = 0.2083333333 -0.375x + x2 = 0.2083333333 The x term is -0.375x. Take half its coefficient (-0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. -0.375x + 0.03515625 + x2 = 0.2083333333 + 0.03515625 Reorder the terms: 0.03515625 + -0.375x + x2 = 0.2083333333 + 0.03515625 Combine like terms: 0.2083333333 + 0.03515625 = 0.2434895833 0.03515625 + -0.375x + x2 = 0.2434895833 Factor a perfect square on the left side: (x + -0.1875)(x + -0.1875) = 0.2434895833 Calculate the square root of the right side: 0.493446637 Break this problem into two subproblems by setting (x + -0.1875) equal to 0.493446637 and -0.493446637.Subproblem 1
x + -0.1875 = 0.493446637 Simplifying x + -0.1875 = 0.493446637 Reorder the terms: -0.1875 + x = 0.493446637 Solving -0.1875 + x = 0.493446637 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + x = 0.493446637 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + x = 0.493446637 + 0.1875 x = 0.493446637 + 0.1875 Combine like terms: 0.493446637 + 0.1875 = 0.680946637 x = 0.680946637 Simplifying x = 0.680946637Subproblem 2
x + -0.1875 = -0.493446637 Simplifying x + -0.1875 = -0.493446637 Reorder the terms: -0.1875 + x = -0.493446637 Solving -0.1875 + x = -0.493446637 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + x = -0.493446637 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + x = -0.493446637 + 0.1875 x = -0.493446637 + 0.1875 Combine like terms: -0.493446637 + 0.1875 = -0.305946637 x = -0.305946637 Simplifying x = -0.305946637Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.680946637, -0.305946637}Solution
x = {0.680946637, -0.305946637}
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