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Simplifying 42x2 + 14x + -10 = 0 Reorder the terms: -10 + 14x + 42x2 = 0 Solving -10 + 14x + 42x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-5 + 7x + 21x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5 + 7x + 21x2)' equal to zero and attempt to solve: Simplifying -5 + 7x + 21x2 = 0 Solving -5 + 7x + 21x2 = 0 Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -0.2380952381 + 0.3333333333x + x2 = 0 Move the constant term to the right: Add '0.2380952381' to each side of the equation. -0.2380952381 + 0.3333333333x + 0.2380952381 + x2 = 0 + 0.2380952381 Reorder the terms: -0.2380952381 + 0.2380952381 + 0.3333333333x + x2 = 0 + 0.2380952381 Combine like terms: -0.2380952381 + 0.2380952381 = 0.0000000000 0.0000000000 + 0.3333333333x + x2 = 0 + 0.2380952381 0.3333333333x + x2 = 0 + 0.2380952381 Combine like terms: 0 + 0.2380952381 = 0.2380952381 0.3333333333x + x2 = 0.2380952381 The x term is 0.3333333333x. Take half its coefficient (0.1666666667). Square it (0.02777777779) and add it to both sides. Add '0.02777777779' to each side of the equation. 0.3333333333x + 0.02777777779 + x2 = 0.2380952381 + 0.02777777779 Reorder the terms: 0.02777777779 + 0.3333333333x + x2 = 0.2380952381 + 0.02777777779 Combine like terms: 0.2380952381 + 0.02777777779 = 0.26587301589 0.02777777779 + 0.3333333333x + x2 = 0.26587301589 Factor a perfect square on the left side: (x + 0.1666666667)(x + 0.1666666667) = 0.26587301589 Calculate the square root of the right side: 0.515628758 Break this problem into two subproblems by setting (x + 0.1666666667) equal to 0.515628758 and -0.515628758.Subproblem 1
x + 0.1666666667 = 0.515628758 Simplifying x + 0.1666666667 = 0.515628758 Reorder the terms: 0.1666666667 + x = 0.515628758 Solving 0.1666666667 + x = 0.515628758 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = 0.515628758 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = 0.515628758 + -0.1666666667 x = 0.515628758 + -0.1666666667 Combine like terms: 0.515628758 + -0.1666666667 = 0.3489620913 x = 0.3489620913 Simplifying x = 0.3489620913Subproblem 2
x + 0.1666666667 = -0.515628758 Simplifying x + 0.1666666667 = -0.515628758 Reorder the terms: 0.1666666667 + x = -0.515628758 Solving 0.1666666667 + x = -0.515628758 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.1666666667' to each side of the equation. 0.1666666667 + -0.1666666667 + x = -0.515628758 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + x = -0.515628758 + -0.1666666667 x = -0.515628758 + -0.1666666667 Combine like terms: -0.515628758 + -0.1666666667 = -0.6822954247 x = -0.6822954247 Simplifying x = -0.6822954247Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.3489620913, -0.6822954247}Solution
x = {0.3489620913, -0.6822954247}
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