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Simplifying 7x2 + -31x + 14 = 0 Reorder the terms: 14 + -31x + 7x2 = 0 Solving 14 + -31x + 7x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 2 + -4.428571429x + x2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + -4.428571429x + -2 + x2 = 0 + -2 Reorder the terms: 2 + -2 + -4.428571429x + x2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -4.428571429x + x2 = 0 + -2 -4.428571429x + x2 = 0 + -2 Combine like terms: 0 + -2 = -2 -4.428571429x + x2 = -2 The x term is -4.428571429x. Take half its coefficient (-2.214285715). Square it (4.903061228) and add it to both sides. Add '4.903061228' to each side of the equation. -4.428571429x + 4.903061228 + x2 = -2 + 4.903061228 Reorder the terms: 4.903061228 + -4.428571429x + x2 = -2 + 4.903061228 Combine like terms: -2 + 4.903061228 = 2.903061228 4.903061228 + -4.428571429x + x2 = 2.903061228 Factor a perfect square on the left side: (x + -2.214285715)(x + -2.214285715) = 2.903061228 Calculate the square root of the right side: 1.703837207 Break this problem into two subproblems by setting (x + -2.214285715) equal to 1.703837207 and -1.703837207.Subproblem 1
x + -2.214285715 = 1.703837207 Simplifying x + -2.214285715 = 1.703837207 Reorder the terms: -2.214285715 + x = 1.703837207 Solving -2.214285715 + x = 1.703837207 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.214285715' to each side of the equation. -2.214285715 + 2.214285715 + x = 1.703837207 + 2.214285715 Combine like terms: -2.214285715 + 2.214285715 = 0.000000000 0.000000000 + x = 1.703837207 + 2.214285715 x = 1.703837207 + 2.214285715 Combine like terms: 1.703837207 + 2.214285715 = 3.918122922 x = 3.918122922 Simplifying x = 3.918122922Subproblem 2
x + -2.214285715 = -1.703837207 Simplifying x + -2.214285715 = -1.703837207 Reorder the terms: -2.214285715 + x = -1.703837207 Solving -2.214285715 + x = -1.703837207 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.214285715' to each side of the equation. -2.214285715 + 2.214285715 + x = -1.703837207 + 2.214285715 Combine like terms: -2.214285715 + 2.214285715 = 0.000000000 0.000000000 + x = -1.703837207 + 2.214285715 x = -1.703837207 + 2.214285715 Combine like terms: -1.703837207 + 2.214285715 = 0.510448508 x = 0.510448508 Simplifying x = 0.510448508Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.918122922, 0.510448508}
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