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Simplifying (5x + -32)(3x + 20) = 180 Reorder the terms: (-32 + 5x)(3x + 20) = 180 Reorder the terms: (-32 + 5x)(20 + 3x) = 180 Multiply (-32 + 5x) * (20 + 3x) (-32(20 + 3x) + 5x * (20 + 3x)) = 180 ((20 * -32 + 3x * -32) + 5x * (20 + 3x)) = 180 ((-640 + -96x) + 5x * (20 + 3x)) = 180 (-640 + -96x + (20 * 5x + 3x * 5x)) = 180 (-640 + -96x + (100x + 15x2)) = 180 Combine like terms: -96x + 100x = 4x (-640 + 4x + 15x2) = 180 Solving -640 + 4x + 15x2 = 180 Solving for variable 'x'. Reorder the terms: -640 + -180 + 4x + 15x2 = 180 + -180 Combine like terms: -640 + -180 = -820 -820 + 4x + 15x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -820 + 4x + 15x2 = 0 Begin completing the square. Divide all terms by 15 the coefficient of the squared term: Divide each side by '15'. -54.66666667 + 0.2666666667x + x2 = 0 Move the constant term to the right: Add '54.66666667' to each side of the equation. -54.66666667 + 0.2666666667x + 54.66666667 + x2 = 0 + 54.66666667 Reorder the terms: -54.66666667 + 54.66666667 + 0.2666666667x + x2 = 0 + 54.66666667 Combine like terms: -54.66666667 + 54.66666667 = 0.00000000 0.00000000 + 0.2666666667x + x2 = 0 + 54.66666667 0.2666666667x + x2 = 0 + 54.66666667 Combine like terms: 0 + 54.66666667 = 54.66666667 0.2666666667x + x2 = 54.66666667 The x term is 0.2666666667x. Take half its coefficient (0.1333333334). Square it (0.01777777780) and add it to both sides. Add '0.01777777780' to each side of the equation. 0.2666666667x + 0.01777777780 + x2 = 54.66666667 + 0.01777777780 Reorder the terms: 0.01777777780 + 0.2666666667x + x2 = 54.66666667 + 0.01777777780 Combine like terms: 54.66666667 + 0.01777777780 = 54.6844444478 0.01777777780 + 0.2666666667x + x2 = 54.6844444478 Factor a perfect square on the left side: (x + 0.1333333334)(x + 0.1333333334) = 54.6844444478 Calculate the square root of the right side: 7.394893133 Break this problem into two subproblems by setting (x + 0.1333333334) equal to 7.394893133 and -7.394893133.Subproblem 1
x + 0.1333333334 = 7.394893133 Simplifying x + 0.1333333334 = 7.394893133 Reorder the terms: 0.1333333334 + x = 7.394893133 Solving 0.1333333334 + x = 7.394893133 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1333333334' to each side of the equation. 0.1333333334 + -0.1333333334 + x = 7.394893133 + -0.1333333334 Combine like terms: 0.1333333334 + -0.1333333334 = 0.0000000000 0.0000000000 + x = 7.394893133 + -0.1333333334 x = 7.394893133 + -0.1333333334 Combine like terms: 7.394893133 + -0.1333333334 = 7.2615597996 x = 7.2615597996 Simplifying x = 7.2615597996Subproblem 2
x + 0.1333333334 = -7.394893133 Simplifying x + 0.1333333334 = -7.394893133 Reorder the terms: 0.1333333334 + x = -7.394893133 Solving 0.1333333334 + x = -7.394893133 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1333333334' to each side of the equation. 0.1333333334 + -0.1333333334 + x = -7.394893133 + -0.1333333334 Combine like terms: 0.1333333334 + -0.1333333334 = 0.0000000000 0.0000000000 + x = -7.394893133 + -0.1333333334 x = -7.394893133 + -0.1333333334 Combine like terms: -7.394893133 + -0.1333333334 = -7.5282264664 x = -7.5282264664 Simplifying x = -7.5282264664Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.2615597996, -7.5282264664}
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