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Simplifying (2x + 5)(3x + 10) = 145 Reorder the terms: (5 + 2x)(3x + 10) = 145 Reorder the terms: (5 + 2x)(10 + 3x) = 145 Multiply (5 + 2x) * (10 + 3x) (5(10 + 3x) + 2x * (10 + 3x)) = 145 ((10 * 5 + 3x * 5) + 2x * (10 + 3x)) = 145 ((50 + 15x) + 2x * (10 + 3x)) = 145 (50 + 15x + (10 * 2x + 3x * 2x)) = 145 (50 + 15x + (20x + 6x2)) = 145 Combine like terms: 15x + 20x = 35x (50 + 35x + 6x2) = 145 Solving 50 + 35x + 6x2 = 145 Solving for variable 'x'. Reorder the terms: 50 + -145 + 35x + 6x2 = 145 + -145 Combine like terms: 50 + -145 = -95 -95 + 35x + 6x2 = 145 + -145 Combine like terms: 145 + -145 = 0 -95 + 35x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -15.83333333 + 5.833333333x + x2 = 0 Move the constant term to the right: Add '15.83333333' to each side of the equation. -15.83333333 + 5.833333333x + 15.83333333 + x2 = 0 + 15.83333333 Reorder the terms: -15.83333333 + 15.83333333 + 5.833333333x + x2 = 0 + 15.83333333 Combine like terms: -15.83333333 + 15.83333333 = 0.00000000 0.00000000 + 5.833333333x + x2 = 0 + 15.83333333 5.833333333x + x2 = 0 + 15.83333333 Combine like terms: 0 + 15.83333333 = 15.83333333 5.833333333x + x2 = 15.83333333 The x term is 5.833333333x. Take half its coefficient (2.916666667). Square it (8.506944446) and add it to both sides. Add '8.506944446' to each side of the equation. 5.833333333x + 8.506944446 + x2 = 15.83333333 + 8.506944446 Reorder the terms: 8.506944446 + 5.833333333x + x2 = 15.83333333 + 8.506944446 Combine like terms: 15.83333333 + 8.506944446 = 24.340277776 8.506944446 + 5.833333333x + x2 = 24.340277776 Factor a perfect square on the left side: (x + 2.916666667)(x + 2.916666667) = 24.340277776 Calculate the square root of the right side: 4.933586705 Break this problem into two subproblems by setting (x + 2.916666667) equal to 4.933586705 and -4.933586705.Subproblem 1
x + 2.916666667 = 4.933586705 Simplifying x + 2.916666667 = 4.933586705 Reorder the terms: 2.916666667 + x = 4.933586705 Solving 2.916666667 + x = 4.933586705 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.916666667' to each side of the equation. 2.916666667 + -2.916666667 + x = 4.933586705 + -2.916666667 Combine like terms: 2.916666667 + -2.916666667 = 0.000000000 0.000000000 + x = 4.933586705 + -2.916666667 x = 4.933586705 + -2.916666667 Combine like terms: 4.933586705 + -2.916666667 = 2.016920038 x = 2.016920038 Simplifying x = 2.016920038Subproblem 2
x + 2.916666667 = -4.933586705 Simplifying x + 2.916666667 = -4.933586705 Reorder the terms: 2.916666667 + x = -4.933586705 Solving 2.916666667 + x = -4.933586705 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.916666667' to each side of the equation. 2.916666667 + -2.916666667 + x = -4.933586705 + -2.916666667 Combine like terms: 2.916666667 + -2.916666667 = 0.000000000 0.000000000 + x = -4.933586705 + -2.916666667 x = -4.933586705 + -2.916666667 Combine like terms: -4.933586705 + -2.916666667 = -7.850253372 x = -7.850253372 Simplifying x = -7.850253372Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.016920038, -7.850253372}
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