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Simplifying 14 = 7x(x) + -19x + 23 Multiply x * x 14 = 7x2 + -19x + 23 Reorder the terms: 14 = 23 + -19x + 7x2 Solving 14 = 23 + -19x + 7x2 Solving for variable 'x'. Combine like terms: 14 + -23 = -9 -9 + 19x + -7x2 = 23 + -19x + 7x2 + -23 + 19x + -7x2 Reorder the terms: -9 + 19x + -7x2 = 23 + -23 + -19x + 19x + 7x2 + -7x2 Combine like terms: 23 + -23 = 0 -9 + 19x + -7x2 = 0 + -19x + 19x + 7x2 + -7x2 -9 + 19x + -7x2 = -19x + 19x + 7x2 + -7x2 Combine like terms: -19x + 19x = 0 -9 + 19x + -7x2 = 0 + 7x2 + -7x2 -9 + 19x + -7x2 = 7x2 + -7x2 Combine like terms: 7x2 + -7x2 = 0 -9 + 19x + -7x2 = 0 Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. 1.285714286 + -2.714285714x + x2 = 0 Move the constant term to the right: Add '-1.285714286' to each side of the equation. 1.285714286 + -2.714285714x + -1.285714286 + x2 = 0 + -1.285714286 Reorder the terms: 1.285714286 + -1.285714286 + -2.714285714x + x2 = 0 + -1.285714286 Combine like terms: 1.285714286 + -1.285714286 = 0.000000000 0.000000000 + -2.714285714x + x2 = 0 + -1.285714286 -2.714285714x + x2 = 0 + -1.285714286 Combine like terms: 0 + -1.285714286 = -1.285714286 -2.714285714x + x2 = -1.285714286 The x term is -2.714285714x. Take half its coefficient (-1.357142857). Square it (1.841836734) and add it to both sides. Add '1.841836734' to each side of the equation. -2.714285714x + 1.841836734 + x2 = -1.285714286 + 1.841836734 Reorder the terms: 1.841836734 + -2.714285714x + x2 = -1.285714286 + 1.841836734 Combine like terms: -1.285714286 + 1.841836734 = 0.556122448 1.841836734 + -2.714285714x + x2 = 0.556122448 Factor a perfect square on the left side: (x + -1.357142857)(x + -1.357142857) = 0.556122448 Calculate the square root of the right side: 0.745736179 Break this problem into two subproblems by setting (x + -1.357142857) equal to 0.745736179 and -0.745736179.Subproblem 1
x + -1.357142857 = 0.745736179 Simplifying x + -1.357142857 = 0.745736179 Reorder the terms: -1.357142857 + x = 0.745736179 Solving -1.357142857 + x = 0.745736179 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.357142857' to each side of the equation. -1.357142857 + 1.357142857 + x = 0.745736179 + 1.357142857 Combine like terms: -1.357142857 + 1.357142857 = 0.000000000 0.000000000 + x = 0.745736179 + 1.357142857 x = 0.745736179 + 1.357142857 Combine like terms: 0.745736179 + 1.357142857 = 2.102879036 x = 2.102879036 Simplifying x = 2.102879036Subproblem 2
x + -1.357142857 = -0.745736179 Simplifying x + -1.357142857 = -0.745736179 Reorder the terms: -1.357142857 + x = -0.745736179 Solving -1.357142857 + x = -0.745736179 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.357142857' to each side of the equation. -1.357142857 + 1.357142857 + x = -0.745736179 + 1.357142857 Combine like terms: -1.357142857 + 1.357142857 = 0.000000000 0.000000000 + x = -0.745736179 + 1.357142857 x = -0.745736179 + 1.357142857 Combine like terms: -0.745736179 + 1.357142857 = 0.611406678 x = 0.611406678 Simplifying x = 0.611406678Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.102879036, 0.611406678}
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