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Simplifying 5x2 + -10x = x2 + 90 Reorder the terms: -10x + 5x2 = x2 + 90 Reorder the terms: -10x + 5x2 = 90 + x2 Solving -10x + 5x2 = 90 + x2 Solving for variable 'x'. Reorder the terms: -90 + -10x + 5x2 + -1x2 = 90 + x2 + -90 + -1x2 Combine like terms: 5x2 + -1x2 = 4x2 -90 + -10x + 4x2 = 90 + x2 + -90 + -1x2 Reorder the terms: -90 + -10x + 4x2 = 90 + -90 + x2 + -1x2 Combine like terms: 90 + -90 = 0 -90 + -10x + 4x2 = 0 + x2 + -1x2 -90 + -10x + 4x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -90 + -10x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-45 + -5x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-45 + -5x + 2x2)' equal to zero and attempt to solve: Simplifying -45 + -5x + 2x2 = 0 Solving -45 + -5x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -22.5 + -2.5x + x2 = 0 Move the constant term to the right: Add '22.5' to each side of the equation. -22.5 + -2.5x + 22.5 + x2 = 0 + 22.5 Reorder the terms: -22.5 + 22.5 + -2.5x + x2 = 0 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + -2.5x + x2 = 0 + 22.5 -2.5x + x2 = 0 + 22.5 Combine like terms: 0 + 22.5 = 22.5 -2.5x + x2 = 22.5 The x term is -2.5x. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5x + 1.5625 + x2 = 22.5 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = 22.5 + 1.5625 Combine like terms: 22.5 + 1.5625 = 24.0625 1.5625 + -2.5x + x2 = 24.0625 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 24.0625 Calculate the square root of the right side: 4.905354218 Break this problem into two subproblems by setting (x + -1.25) equal to 4.905354218 and -4.905354218.Subproblem 1
x + -1.25 = 4.905354218 Simplifying x + -1.25 = 4.905354218 Reorder the terms: -1.25 + x = 4.905354218 Solving -1.25 + x = 4.905354218 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 4.905354218 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 4.905354218 + 1.25 x = 4.905354218 + 1.25 Combine like terms: 4.905354218 + 1.25 = 6.155354218 x = 6.155354218 Simplifying x = 6.155354218Subproblem 2
x + -1.25 = -4.905354218 Simplifying x + -1.25 = -4.905354218 Reorder the terms: -1.25 + x = -4.905354218 Solving -1.25 + x = -4.905354218 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -4.905354218 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -4.905354218 + 1.25 x = -4.905354218 + 1.25 Combine like terms: -4.905354218 + 1.25 = -3.655354218 x = -3.655354218 Simplifying x = -3.655354218Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.155354218, -3.655354218}Solution
x = {6.155354218, -3.655354218}
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