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Simplifying (8x + 4)(2x + -6) = (7x + -3)(5 + 9x) Reorder the terms: (4 + 8x)(2x + -6) = (7x + -3)(5 + 9x) Reorder the terms: (4 + 8x)(-6 + 2x) = (7x + -3)(5 + 9x) Multiply (4 + 8x) * (-6 + 2x) (4(-6 + 2x) + 8x * (-6 + 2x)) = (7x + -3)(5 + 9x) ((-6 * 4 + 2x * 4) + 8x * (-6 + 2x)) = (7x + -3)(5 + 9x) ((-24 + 8x) + 8x * (-6 + 2x)) = (7x + -3)(5 + 9x) (-24 + 8x + (-6 * 8x + 2x * 8x)) = (7x + -3)(5 + 9x) (-24 + 8x + (-48x + 16x2)) = (7x + -3)(5 + 9x) Combine like terms: 8x + -48x = -40x (-24 + -40x + 16x2) = (7x + -3)(5 + 9x) Reorder the terms: -24 + -40x + 16x2 = (-3 + 7x)(5 + 9x) Multiply (-3 + 7x) * (5 + 9x) -24 + -40x + 16x2 = (-3(5 + 9x) + 7x * (5 + 9x)) -24 + -40x + 16x2 = ((5 * -3 + 9x * -3) + 7x * (5 + 9x)) -24 + -40x + 16x2 = ((-15 + -27x) + 7x * (5 + 9x)) -24 + -40x + 16x2 = (-15 + -27x + (5 * 7x + 9x * 7x)) -24 + -40x + 16x2 = (-15 + -27x + (35x + 63x2)) Combine like terms: -27x + 35x = 8x -24 + -40x + 16x2 = (-15 + 8x + 63x2) Solving -24 + -40x + 16x2 = -15 + 8x + 63x2 Solving for variable 'x'. Reorder the terms: -24 + 15 + -40x + -8x + 16x2 + -63x2 = -15 + 8x + 63x2 + 15 + -8x + -63x2 Combine like terms: -24 + 15 = -9 -9 + -40x + -8x + 16x2 + -63x2 = -15 + 8x + 63x2 + 15 + -8x + -63x2 Combine like terms: -40x + -8x = -48x -9 + -48x + 16x2 + -63x2 = -15 + 8x + 63x2 + 15 + -8x + -63x2 Combine like terms: 16x2 + -63x2 = -47x2 -9 + -48x + -47x2 = -15 + 8x + 63x2 + 15 + -8x + -63x2 Reorder the terms: -9 + -48x + -47x2 = -15 + 15 + 8x + -8x + 63x2 + -63x2 Combine like terms: -15 + 15 = 0 -9 + -48x + -47x2 = 0 + 8x + -8x + 63x2 + -63x2 -9 + -48x + -47x2 = 8x + -8x + 63x2 + -63x2 Combine like terms: 8x + -8x = 0 -9 + -48x + -47x2 = 0 + 63x2 + -63x2 -9 + -48x + -47x2 = 63x2 + -63x2 Combine like terms: 63x2 + -63x2 = 0 -9 + -48x + -47x2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(9 + 48x + 47x2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(9 + 48x + 47x2)' equal to zero and attempt to solve: Simplifying 9 + 48x + 47x2 = 0 Solving 9 + 48x + 47x2 = 0 Begin completing the square. Divide all terms by 47 the coefficient of the squared term: Divide each side by '47'. 0.1914893617 + 1.021276596x + x2 = 0 Move the constant term to the right: Add '-0.1914893617' to each side of the equation. 0.1914893617 + 1.021276596x + -0.1914893617 + x2 = 0 + -0.1914893617 Reorder the terms: 0.1914893617 + -0.1914893617 + 1.021276596x + x2 = 0 + -0.1914893617 Combine like terms: 0.1914893617 + -0.1914893617 = 0.0000000000 0.0000000000 + 1.021276596x + x2 = 0 + -0.1914893617 1.021276596x + x2 = 0 + -0.1914893617 Combine like terms: 0 + -0.1914893617 = -0.1914893617 1.021276596x + x2 = -0.1914893617 The x term is 1.021276596x. Take half its coefficient (0.510638298). Square it (0.2607514714) and add it to both sides. Add '0.2607514714' to each side of the equation. 1.021276596x + 0.2607514714 + x2 = -0.1914893617 + 0.2607514714 Reorder the terms: 0.2607514714 + 1.021276596x + x2 = -0.1914893617 + 0.2607514714 Combine like terms: -0.1914893617 + 0.2607514714 = 0.0692621097 0.2607514714 + 1.021276596x + x2 = 0.0692621097 Factor a perfect square on the left side: (x + 0.510638298)(x + 0.510638298) = 0.0692621097 Calculate the square root of the right side: 0.263176955 Break this problem into two subproblems by setting (x + 0.510638298) equal to 0.263176955 and -0.263176955.Subproblem 1
x + 0.510638298 = 0.263176955 Simplifying x + 0.510638298 = 0.263176955 Reorder the terms: 0.510638298 + x = 0.263176955 Solving 0.510638298 + x = 0.263176955 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.510638298' to each side of the equation. 0.510638298 + -0.510638298 + x = 0.263176955 + -0.510638298 Combine like terms: 0.510638298 + -0.510638298 = 0.000000000 0.000000000 + x = 0.263176955 + -0.510638298 x = 0.263176955 + -0.510638298 Combine like terms: 0.263176955 + -0.510638298 = -0.247461343 x = -0.247461343 Simplifying x = -0.247461343Subproblem 2
x + 0.510638298 = -0.263176955 Simplifying x + 0.510638298 = -0.263176955 Reorder the terms: 0.510638298 + x = -0.263176955 Solving 0.510638298 + x = -0.263176955 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.510638298' to each side of the equation. 0.510638298 + -0.510638298 + x = -0.263176955 + -0.510638298 Combine like terms: 0.510638298 + -0.510638298 = 0.000000000 0.000000000 + x = -0.263176955 + -0.510638298 x = -0.263176955 + -0.510638298 Combine like terms: -0.263176955 + -0.510638298 = -0.773815253 x = -0.773815253 Simplifying x = -0.773815253Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.247461343, -0.773815253}Solution
x = {-0.247461343, -0.773815253}
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