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Simplifying 10x2 + 2x + 1 = 501 Reorder the terms: 1 + 2x + 10x2 = 501 Solving 1 + 2x + 10x2 = 501 Solving for variable 'x'. Reorder the terms: 1 + -501 + 2x + 10x2 = 501 + -501 Combine like terms: 1 + -501 = -500 -500 + 2x + 10x2 = 501 + -501 Combine like terms: 501 + -501 = 0 -500 + 2x + 10x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-250 + x + 5x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-250 + x + 5x2)' equal to zero and attempt to solve: Simplifying -250 + x + 5x2 = 0 Solving -250 + x + 5x2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -50 + 0.2x + x2 = 0 Move the constant term to the right: Add '50' to each side of the equation. -50 + 0.2x + 50 + x2 = 0 + 50 Reorder the terms: -50 + 50 + 0.2x + x2 = 0 + 50 Combine like terms: -50 + 50 = 0 0 + 0.2x + x2 = 0 + 50 0.2x + x2 = 0 + 50 Combine like terms: 0 + 50 = 50 0.2x + x2 = 50 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.2x + 0.25 + x2 = 50 + 0.25 Reorder the terms: 0.25 + 0.2x + x2 = 50 + 0.25 Combine like terms: 50 + 0.25 = 50.25 0.25 + 0.2x + x2 = 50.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 50.25 Calculate the square root of the right side: 7.088723439 Break this problem into two subproblems by setting (x + 0.5) equal to 7.088723439 and -7.088723439.Subproblem 1
x + 0.5 = 7.088723439 Simplifying x + 0.5 = 7.088723439 Reorder the terms: 0.5 + x = 7.088723439 Solving 0.5 + x = 7.088723439 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 7.088723439 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 7.088723439 + -0.5 x = 7.088723439 + -0.5 Combine like terms: 7.088723439 + -0.5 = 6.588723439 x = 6.588723439 Simplifying x = 6.588723439Subproblem 2
x + 0.5 = -7.088723439 Simplifying x + 0.5 = -7.088723439 Reorder the terms: 0.5 + x = -7.088723439 Solving 0.5 + x = -7.088723439 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -7.088723439 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -7.088723439 + -0.5 x = -7.088723439 + -0.5 Combine like terms: -7.088723439 + -0.5 = -7.588723439 x = -7.588723439 Simplifying x = -7.588723439Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.588723439, -7.588723439}Solution
x = {6.588723439, -7.588723439}
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