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Simplifying 0 = -7x2 + -11x + 3 Reorder the terms: 0 = 3 + -11x + -7x2 Solving 0 = 3 + -11x + -7x2 Solving for variable 'x'. Combine like terms: 0 + -3 = -3 -3 + 11x + 7x2 = 3 + -11x + -7x2 + -3 + 11x + 7x2 Reorder the terms: -3 + 11x + 7x2 = 3 + -3 + -11x + 11x + -7x2 + 7x2 Combine like terms: 3 + -3 = 0 -3 + 11x + 7x2 = 0 + -11x + 11x + -7x2 + 7x2 -3 + 11x + 7x2 = -11x + 11x + -7x2 + 7x2 Combine like terms: -11x + 11x = 0 -3 + 11x + 7x2 = 0 + -7x2 + 7x2 -3 + 11x + 7x2 = -7x2 + 7x2 Combine like terms: -7x2 + 7x2 = 0 -3 + 11x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.4285714286 + 1.571428571x + x2 = 0 Move the constant term to the right: Add '0.4285714286' to each side of the equation. -0.4285714286 + 1.571428571x + 0.4285714286 + x2 = 0 + 0.4285714286 Reorder the terms: -0.4285714286 + 0.4285714286 + 1.571428571x + x2 = 0 + 0.4285714286 Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000 0.0000000000 + 1.571428571x + x2 = 0 + 0.4285714286 1.571428571x + x2 = 0 + 0.4285714286 Combine like terms: 0 + 0.4285714286 = 0.4285714286 1.571428571x + x2 = 0.4285714286 The x term is 1.571428571x. Take half its coefficient (0.7857142855). Square it (0.6173469384) and add it to both sides. Add '0.6173469384' to each side of the equation. 1.571428571x + 0.6173469384 + x2 = 0.4285714286 + 0.6173469384 Reorder the terms: 0.6173469384 + 1.571428571x + x2 = 0.4285714286 + 0.6173469384 Combine like terms: 0.4285714286 + 0.6173469384 = 1.045918367 0.6173469384 + 1.571428571x + x2 = 1.045918367 Factor a perfect square on the left side: (x + 0.7857142855)(x + 0.7857142855) = 1.045918367 Calculate the square root of the right side: 1.022701504 Break this problem into two subproblems by setting (x + 0.7857142855) equal to 1.022701504 and -1.022701504.Subproblem 1
x + 0.7857142855 = 1.022701504 Simplifying x + 0.7857142855 = 1.022701504 Reorder the terms: 0.7857142855 + x = 1.022701504 Solving 0.7857142855 + x = 1.022701504 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7857142855' to each side of the equation. 0.7857142855 + -0.7857142855 + x = 1.022701504 + -0.7857142855 Combine like terms: 0.7857142855 + -0.7857142855 = 0.0000000000 0.0000000000 + x = 1.022701504 + -0.7857142855 x = 1.022701504 + -0.7857142855 Combine like terms: 1.022701504 + -0.7857142855 = 0.2369872185 x = 0.2369872185 Simplifying x = 0.2369872185Subproblem 2
x + 0.7857142855 = -1.022701504 Simplifying x + 0.7857142855 = -1.022701504 Reorder the terms: 0.7857142855 + x = -1.022701504 Solving 0.7857142855 + x = -1.022701504 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7857142855' to each side of the equation. 0.7857142855 + -0.7857142855 + x = -1.022701504 + -0.7857142855 Combine like terms: 0.7857142855 + -0.7857142855 = 0.0000000000 0.0000000000 + x = -1.022701504 + -0.7857142855 x = -1.022701504 + -0.7857142855 Combine like terms: -1.022701504 + -0.7857142855 = -1.8084157895 x = -1.8084157895 Simplifying x = -1.8084157895Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.2369872185, -1.8084157895}
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