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Simplifying 78x2 + -16x + -28 = 0 Reorder the terms: -28 + -16x + 78x2 = 0 Solving -28 + -16x + 78x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-14 + -8x + 39x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-14 + -8x + 39x2)' equal to zero and attempt to solve: Simplifying -14 + -8x + 39x2 = 0 Solving -14 + -8x + 39x2 = 0 Begin completing the square. Divide all terms by 39 the coefficient of the squared term: Divide each side by '39'. -0.358974359 + -0.2051282051x + x2 = 0 Move the constant term to the right: Add '0.358974359' to each side of the equation. -0.358974359 + -0.2051282051x + 0.358974359 + x2 = 0 + 0.358974359 Reorder the terms: -0.358974359 + 0.358974359 + -0.2051282051x + x2 = 0 + 0.358974359 Combine like terms: -0.358974359 + 0.358974359 = 0.000000000 0.000000000 + -0.2051282051x + x2 = 0 + 0.358974359 -0.2051282051x + x2 = 0 + 0.358974359 Combine like terms: 0 + 0.358974359 = 0.358974359 -0.2051282051x + x2 = 0.358974359 The x term is -0.2051282051x. Take half its coefficient (-0.1025641026). Square it (0.01051939514) and add it to both sides. Add '0.01051939514' to each side of the equation. -0.2051282051x + 0.01051939514 + x2 = 0.358974359 + 0.01051939514 Reorder the terms: 0.01051939514 + -0.2051282051x + x2 = 0.358974359 + 0.01051939514 Combine like terms: 0.358974359 + 0.01051939514 = 0.36949375414 0.01051939514 + -0.2051282051x + x2 = 0.36949375414 Factor a perfect square on the left side: (x + -0.1025641026)(x + -0.1025641026) = 0.36949375414 Calculate the square root of the right side: 0.607859979 Break this problem into two subproblems by setting (x + -0.1025641026) equal to 0.607859979 and -0.607859979.Subproblem 1
x + -0.1025641026 = 0.607859979 Simplifying x + -0.1025641026 = 0.607859979 Reorder the terms: -0.1025641026 + x = 0.607859979 Solving -0.1025641026 + x = 0.607859979 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1025641026' to each side of the equation. -0.1025641026 + 0.1025641026 + x = 0.607859979 + 0.1025641026 Combine like terms: -0.1025641026 + 0.1025641026 = 0.0000000000 0.0000000000 + x = 0.607859979 + 0.1025641026 x = 0.607859979 + 0.1025641026 Combine like terms: 0.607859979 + 0.1025641026 = 0.7104240816 x = 0.7104240816 Simplifying x = 0.7104240816Subproblem 2
x + -0.1025641026 = -0.607859979 Simplifying x + -0.1025641026 = -0.607859979 Reorder the terms: -0.1025641026 + x = -0.607859979 Solving -0.1025641026 + x = -0.607859979 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1025641026' to each side of the equation. -0.1025641026 + 0.1025641026 + x = -0.607859979 + 0.1025641026 Combine like terms: -0.1025641026 + 0.1025641026 = 0.0000000000 0.0000000000 + x = -0.607859979 + 0.1025641026 x = -0.607859979 + 0.1025641026 Combine like terms: -0.607859979 + 0.1025641026 = -0.5052958764 x = -0.5052958764 Simplifying x = -0.5052958764Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.7104240816, -0.5052958764}Solution
x = {0.7104240816, -0.5052958764}
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