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Simplifying 12(x + -6)(x + 3) + -5(x + 7)(x + -4) = 60(x + -5)(x + -10) Reorder the terms: 12(-6 + x)(x + 3) + -5(x + 7)(x + -4) = 60(x + -5)(x + -10) Reorder the terms: 12(-6 + x)(3 + x) + -5(x + 7)(x + -4) = 60(x + -5)(x + -10) Multiply (-6 + x) * (3 + x) 12(-6(3 + x) + x(3 + x)) + -5(x + 7)(x + -4) = 60(x + -5)(x + -10) 12((3 * -6 + x * -6) + x(3 + x)) + -5(x + 7)(x + -4) = 60(x + -5)(x + -10) 12((-18 + -6x) + x(3 + x)) + -5(x + 7)(x + -4) = 60(x + -5)(x + -10) 12(-18 + -6x + (3 * x + x * x)) + -5(x + 7)(x + -4) = 60(x + -5)(x + -10) 12(-18 + -6x + (3x + x2)) + -5(x + 7)(x + -4) = 60(x + -5)(x + -10) Combine like terms: -6x + 3x = -3x 12(-18 + -3x + x2) + -5(x + 7)(x + -4) = 60(x + -5)(x + -10) (-18 * 12 + -3x * 12 + x2 * 12) + -5(x + 7)(x + -4) = 60(x + -5)(x + -10) (-216 + -36x + 12x2) + -5(x + 7)(x + -4) = 60(x + -5)(x + -10) Reorder the terms: -216 + -36x + 12x2 + -5(7 + x)(x + -4) = 60(x + -5)(x + -10) Reorder the terms: -216 + -36x + 12x2 + -5(7 + x)(-4 + x) = 60(x + -5)(x + -10) Multiply (7 + x) * (-4 + x) -216 + -36x + 12x2 + -5(7(-4 + x) + x(-4 + x)) = 60(x + -5)(x + -10) -216 + -36x + 12x2 + -5((-4 * 7 + x * 7) + x(-4 + x)) = 60(x + -5)(x + -10) -216 + -36x + 12x2 + -5((-28 + 7x) + x(-4 + x)) = 60(x + -5)(x + -10) -216 + -36x + 12x2 + -5(-28 + 7x + (-4 * x + x * x)) = 60(x + -5)(x + -10) -216 + -36x + 12x2 + -5(-28 + 7x + (-4x + x2)) = 60(x + -5)(x + -10) Combine like terms: 7x + -4x = 3x -216 + -36x + 12x2 + -5(-28 + 3x + x2) = 60(x + -5)(x + -10) -216 + -36x + 12x2 + (-28 * -5 + 3x * -5 + x2 * -5) = 60(x + -5)(x + -10) -216 + -36x + 12x2 + (140 + -15x + -5x2) = 60(x + -5)(x + -10) Reorder the terms: -216 + 140 + -36x + -15x + 12x2 + -5x2 = 60(x + -5)(x + -10) Combine like terms: -216 + 140 = -76 -76 + -36x + -15x + 12x2 + -5x2 = 60(x + -5)(x + -10) Combine like terms: -36x + -15x = -51x -76 + -51x + 12x2 + -5x2 = 60(x + -5)(x + -10) Combine like terms: 12x2 + -5x2 = 7x2 -76 + -51x + 7x2 = 60(x + -5)(x + -10) Reorder the terms: -76 + -51x + 7x2 = 60(-5 + x)(x + -10) Reorder the terms: -76 + -51x + 7x2 = 60(-5 + x)(-10 + x) Multiply (-5 + x) * (-10 + x) -76 + -51x + 7x2 = 60(-5(-10 + x) + x(-10 + x)) -76 + -51x + 7x2 = 60((-10 * -5 + x * -5) + x(-10 + x)) -76 + -51x + 7x2 = 60((50 + -5x) + x(-10 + x)) -76 + -51x + 7x2 = 60(50 + -5x + (-10 * x + x * x)) -76 + -51x + 7x2 = 60(50 + -5x + (-10x + x2)) Combine like terms: -5x + -10x = -15x -76 + -51x + 7x2 = 60(50 + -15x + x2) -76 + -51x + 7x2 = (50 * 60 + -15x * 60 + x2 * 60) -76 + -51x + 7x2 = (3000 + -900x + 60x2) Solving -76 + -51x + 7x2 = 3000 + -900x + 60x2 Solving for variable 'x'. Reorder the terms: -76 + -3000 + -51x + 900x + 7x2 + -60x2 = 3000 + -900x + 60x2 + -3000 + 900x + -60x2 Combine like terms: -76 + -3000 = -3076 -3076 + -51x + 900x + 7x2 + -60x2 = 3000 + -900x + 60x2 + -3000 + 900x + -60x2 Combine like terms: -51x + 900x = 849x -3076 + 849x + 7x2 + -60x2 = 3000 + -900x + 60x2 + -3000 + 900x + -60x2 Combine like terms: 7x2 + -60x2 = -53x2 -3076 + 849x + -53x2 = 3000 + -900x + 60x2 + -3000 + 900x + -60x2 Reorder the terms: -3076 + 849x + -53x2 = 3000 + -3000 + -900x + 900x + 60x2 + -60x2 Combine like terms: 3000 + -3000 = 0 -3076 + 849x + -53x2 = 0 + -900x + 900x + 60x2 + -60x2 -3076 + 849x + -53x2 = -900x + 900x + 60x2 + -60x2 Combine like terms: -900x + 900x = 0 -3076 + 849x + -53x2 = 0 + 60x2 + -60x2 -3076 + 849x + -53x2 = 60x2 + -60x2 Combine like terms: 60x2 + -60x2 = 0 -3076 + 849x + -53x2 = 0 Begin completing the square. Divide all terms by -53 the coefficient of the squared term: Divide each side by '-53'. 58.03773585 + -16.01886792x + x2 = 0 Move the constant term to the right: Add '-58.03773585' to each side of the equation. 58.03773585 + -16.01886792x + -58.03773585 + x2 = 0 + -58.03773585 Reorder the terms: 58.03773585 + -58.03773585 + -16.01886792x + x2 = 0 + -58.03773585 Combine like terms: 58.03773585 + -58.03773585 = 0.00000000 0.00000000 + -16.01886792x + x2 = 0 + -58.03773585 -16.01886792x + x2 = 0 + -58.03773585 Combine like terms: 0 + -58.03773585 = -58.03773585 -16.01886792x + x2 = -58.03773585 The x term is -16.01886792x. Take half its coefficient (-8.00943396). Square it (64.15103236) and add it to both sides. Add '64.15103236' to each side of the equation. -16.01886792x + 64.15103236 + x2 = -58.03773585 + 64.15103236 Reorder the terms: 64.15103236 + -16.01886792x + x2 = -58.03773585 + 64.15103236 Combine like terms: -58.03773585 + 64.15103236 = 6.11329651 64.15103236 + -16.01886792x + x2 = 6.11329651 Factor a perfect square on the left side: (x + -8.00943396)(x + -8.00943396) = 6.11329651 Calculate the square root of the right side: 2.472508142 Break this problem into two subproblems by setting (x + -8.00943396) equal to 2.472508142 and -2.472508142.Subproblem 1
x + -8.00943396 = 2.472508142 Simplifying x + -8.00943396 = 2.472508142 Reorder the terms: -8.00943396 + x = 2.472508142 Solving -8.00943396 + x = 2.472508142 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8.00943396' to each side of the equation. -8.00943396 + 8.00943396 + x = 2.472508142 + 8.00943396 Combine like terms: -8.00943396 + 8.00943396 = 0.00000000 0.00000000 + x = 2.472508142 + 8.00943396 x = 2.472508142 + 8.00943396 Combine like terms: 2.472508142 + 8.00943396 = 10.481942102 x = 10.481942102 Simplifying x = 10.481942102Subproblem 2
x + -8.00943396 = -2.472508142 Simplifying x + -8.00943396 = -2.472508142 Reorder the terms: -8.00943396 + x = -2.472508142 Solving -8.00943396 + x = -2.472508142 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '8.00943396' to each side of the equation. -8.00943396 + 8.00943396 + x = -2.472508142 + 8.00943396 Combine like terms: -8.00943396 + 8.00943396 = 0.00000000 0.00000000 + x = -2.472508142 + 8.00943396 x = -2.472508142 + 8.00943396 Combine like terms: -2.472508142 + 8.00943396 = 5.536925818 x = 5.536925818 Simplifying x = 5.536925818Solution
The solution to the problem is based on the solutions from the subproblems. x = {10.481942102, 5.536925818}
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