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Simplifying 70x2 + 140x + -90 = 0 Reorder the terms: -90 + 140x + 70x2 = 0 Solving -90 + 140x + 70x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '10'. 10(-9 + 14x + 7x2) = 0 Ignore the factor 10.Subproblem 1
Set the factor '(-9 + 14x + 7x2)' equal to zero and attempt to solve: Simplifying -9 + 14x + 7x2 = 0 Solving -9 + 14x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -1.285714286 + 2x + x2 = 0 Move the constant term to the right: Add '1.285714286' to each side of the equation. -1.285714286 + 2x + 1.285714286 + x2 = 0 + 1.285714286 Reorder the terms: -1.285714286 + 1.285714286 + 2x + x2 = 0 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + 2x + x2 = 0 + 1.285714286 2x + x2 = 0 + 1.285714286 Combine like terms: 0 + 1.285714286 = 1.285714286 2x + x2 = 1.285714286 The x term is 2x. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2x + 1 + x2 = 1.285714286 + 1 Reorder the terms: 1 + 2x + x2 = 1.285714286 + 1 Combine like terms: 1.285714286 + 1 = 2.285714286 1 + 2x + x2 = 2.285714286 Factor a perfect square on the left side: (x + 1)(x + 1) = 2.285714286 Calculate the square root of the right side: 1.511857892 Break this problem into two subproblems by setting (x + 1) equal to 1.511857892 and -1.511857892.Subproblem 1
x + 1 = 1.511857892 Simplifying x + 1 = 1.511857892 Reorder the terms: 1 + x = 1.511857892 Solving 1 + x = 1.511857892 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = 1.511857892 + -1 Combine like terms: 1 + -1 = 0 0 + x = 1.511857892 + -1 x = 1.511857892 + -1 Combine like terms: 1.511857892 + -1 = 0.511857892 x = 0.511857892 Simplifying x = 0.511857892Subproblem 2
x + 1 = -1.511857892 Simplifying x + 1 = -1.511857892 Reorder the terms: 1 + x = -1.511857892 Solving 1 + x = -1.511857892 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + x = -1.511857892 + -1 Combine like terms: 1 + -1 = 0 0 + x = -1.511857892 + -1 x = -1.511857892 + -1 Combine like terms: -1.511857892 + -1 = -2.511857892 x = -2.511857892 Simplifying x = -2.511857892Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.511857892, -2.511857892}Solution
x = {0.511857892, -2.511857892}
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