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Simplifying (3x + 20)(5x + -8) = 180 Reorder the terms: (20 + 3x)(5x + -8) = 180 Reorder the terms: (20 + 3x)(-8 + 5x) = 180 Multiply (20 + 3x) * (-8 + 5x) (20(-8 + 5x) + 3x * (-8 + 5x)) = 180 ((-8 * 20 + 5x * 20) + 3x * (-8 + 5x)) = 180 ((-160 + 100x) + 3x * (-8 + 5x)) = 180 (-160 + 100x + (-8 * 3x + 5x * 3x)) = 180 (-160 + 100x + (-24x + 15x2)) = 180 Combine like terms: 100x + -24x = 76x (-160 + 76x + 15x2) = 180 Solving -160 + 76x + 15x2 = 180 Solving for variable 'x'. Reorder the terms: -160 + -180 + 76x + 15x2 = 180 + -180 Combine like terms: -160 + -180 = -340 -340 + 76x + 15x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -340 + 76x + 15x2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -22.66666667 + 5.066666667x + x2 = 0 Move the constant term to the right: Add '22.66666667' to each side of the equation. -22.66666667 + 5.066666667x + 22.66666667 + x2 = 0 + 22.66666667 Reorder the terms: -22.66666667 + 22.66666667 + 5.066666667x + x2 = 0 + 22.66666667 Combine like terms: -22.66666667 + 22.66666667 = 0.00000000 0.00000000 + 5.066666667x + x2 = 0 + 22.66666667 5.066666667x + x2 = 0 + 22.66666667 Combine like terms: 0 + 22.66666667 = 22.66666667 5.066666667x + x2 = 22.66666667 The x term is 5.066666667x. Take half its coefficient (2.533333334). Square it (6.417777781) and add it to both sides. Add '6.417777781' to each side of the equation. 5.066666667x + 6.417777781 + x2 = 22.66666667 + 6.417777781 Reorder the terms: 6.417777781 + 5.066666667x + x2 = 22.66666667 + 6.417777781 Combine like terms: 22.66666667 + 6.417777781 = 29.084444451 6.417777781 + 5.066666667x + x2 = 29.084444451 Factor a perfect square on the left side: (x + 2.533333334)(x + 2.533333334) = 29.084444451 Calculate the square root of the right side: 5.392999578 Break this problem into two subproblems by setting (x + 2.533333334) equal to 5.392999578 and -5.392999578.Subproblem 1
x + 2.533333334 = 5.392999578 Simplifying x + 2.533333334 = 5.392999578 Reorder the terms: 2.533333334 + x = 5.392999578 Solving 2.533333334 + x = 5.392999578 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.533333334' to each side of the equation. 2.533333334 + -2.533333334 + x = 5.392999578 + -2.533333334 Combine like terms: 2.533333334 + -2.533333334 = 0.000000000 0.000000000 + x = 5.392999578 + -2.533333334 x = 5.392999578 + -2.533333334 Combine like terms: 5.392999578 + -2.533333334 = 2.859666244 x = 2.859666244 Simplifying x = 2.859666244Subproblem 2
x + 2.533333334 = -5.392999578 Simplifying x + 2.533333334 = -5.392999578 Reorder the terms: 2.533333334 + x = -5.392999578 Solving 2.533333334 + x = -5.392999578 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.533333334' to each side of the equation. 2.533333334 + -2.533333334 + x = -5.392999578 + -2.533333334 Combine like terms: 2.533333334 + -2.533333334 = 0.000000000 0.000000000 + x = -5.392999578 + -2.533333334 x = -5.392999578 + -2.533333334 Combine like terms: -5.392999578 + -2.533333334 = -7.926332912 x = -7.926332912 Simplifying x = -7.926332912Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.859666244, -7.926332912}
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