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Simplifying (X + 10) * 45(4X + -19) * 45 + (2X) * 90 = 0 Reorder the terms: (10 + X) * 45(4X + -19) * 45 + (2X) * 90 = 0 Reorder the terms: (10 + X) * 45(-19 + 4X) * 45 + (2X) * 90 = 0 Reorder the terms for easier multiplication: 45 * 45(10 + X)(-19 + 4X) + (2X) * 90 = 0 Multiply 45 * 45 2025(10 + X)(-19 + 4X) + (2X) * 90 = 0 Multiply (10 + X) * (-19 + 4X) 2025(10(-19 + 4X) + X(-19 + 4X)) + (2X) * 90 = 0 2025((-19 * 10 + 4X * 10) + X(-19 + 4X)) + (2X) * 90 = 0 2025((-190 + 40X) + X(-19 + 4X)) + (2X) * 90 = 0 2025(-190 + 40X + (-19 * X + 4X * X)) + (2X) * 90 = 0 2025(-190 + 40X + (-19X + 4X2)) + (2X) * 90 = 0 Combine like terms: 40X + -19X = 21X 2025(-190 + 21X + 4X2) + (2X) * 90 = 0 (-190 * 2025 + 21X * 2025 + 4X2 * 2025) + (2X) * 90 = 0 (-384750 + 42525X + 8100X2) + (2X) * 90 = 0 Remove parenthesis around (2X) -384750 + 42525X + 8100X2 + 2X * 90 = 0 Reorder the terms for easier multiplication: -384750 + 42525X + 8100X2 + 2 * 90X = 0 Multiply 2 * 90 -384750 + 42525X + 8100X2 + 180X = 0 Reorder the terms: -384750 + 42525X + 180X + 8100X2 = 0 Combine like terms: 42525X + 180X = 42705X -384750 + 42705X + 8100X2 = 0 Solving -384750 + 42705X + 8100X2 = 0 Solving for variable 'X'. Factor out the Greatest Common Factor (GCF), '45'. 45(-8550 + 949X + 180X2) = 0 Ignore the factor 45.Subproblem 1
Set the factor '(-8550 + 949X + 180X2)' equal to zero and attempt to solve: Simplifying -8550 + 949X + 180X2 = 0 Solving -8550 + 949X + 180X2 = 0 Begin completing the square. Divide all terms by 180 the coefficient of the squared term: Divide each side by '180'. -47.5 + 5.272222222X + X2 = 0.0 Move the constant term to the right: Add '47.5' to each side of the equation. -47.5 + 5.272222222X + 47.5 + X2 = 0.0 + 47.5 Reorder the terms: -47.5 + 47.5 + 5.272222222X + X2 = 0.0 + 47.5 Combine like terms: -47.5 + 47.5 = 0.0 0.0 + 5.272222222X + X2 = 0.0 + 47.5 5.272222222X + X2 = 0.0 + 47.5 Combine like terms: 0.0 + 47.5 = 47.5 5.272222222X + X2 = 47.5 The X term is 5.272222222X. Take half its coefficient (2.636111111). Square it (6.949081790) and add it to both sides. Add '6.949081790' to each side of the equation. 5.272222222X + 6.949081790 + X2 = 47.5 + 6.949081790 Reorder the terms: 6.949081790 + 5.272222222X + X2 = 47.5 + 6.949081790 Combine like terms: 47.5 + 6.949081790 = 54.44908179 6.949081790 + 5.272222222X + X2 = 54.44908179 Factor a perfect square on the left side: (X + 2.636111111)(X + 2.636111111) = 54.44908179 Calculate the square root of the right side: 7.378962108 Break this problem into two subproblems by setting (X + 2.636111111) equal to 7.378962108 and -7.378962108.Subproblem 1
X + 2.636111111 = 7.378962108 Simplifying X + 2.636111111 = 7.378962108 Reorder the terms: 2.636111111 + X = 7.378962108 Solving 2.636111111 + X = 7.378962108 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-2.636111111' to each side of the equation. 2.636111111 + -2.636111111 + X = 7.378962108 + -2.636111111 Combine like terms: 2.636111111 + -2.636111111 = 0.000000000 0.000000000 + X = 7.378962108 + -2.636111111 X = 7.378962108 + -2.636111111 Combine like terms: 7.378962108 + -2.636111111 = 4.742850997 X = 4.742850997 Simplifying X = 4.742850997Subproblem 2
X + 2.636111111 = -7.378962108 Simplifying X + 2.636111111 = -7.378962108 Reorder the terms: 2.636111111 + X = -7.378962108 Solving 2.636111111 + X = -7.378962108 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-2.636111111' to each side of the equation. 2.636111111 + -2.636111111 + X = -7.378962108 + -2.636111111 Combine like terms: 2.636111111 + -2.636111111 = 0.000000000 0.000000000 + X = -7.378962108 + -2.636111111 X = -7.378962108 + -2.636111111 Combine like terms: -7.378962108 + -2.636111111 = -10.015073219 X = -10.015073219 Simplifying X = -10.015073219Solution
The solution to the problem is based on the solutions from the subproblems. X = {4.742850997, -10.015073219}Solution
X = {4.742850997, -10.015073219}
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