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Simplifying (3x2 + 7x + -5) + 3(-7x2 + 4x + 3) = 0 Reorder the terms: (-5 + 7x + 3x2) + 3(-7x2 + 4x + 3) = 0 Remove parenthesis around (-5 + 7x + 3x2) -5 + 7x + 3x2 + 3(-7x2 + 4x + 3) = 0 Reorder the terms: -5 + 7x + 3x2 + 3(3 + 4x + -7x2) = 0 -5 + 7x + 3x2 + (3 * 3 + 4x * 3 + -7x2 * 3) = 0 -5 + 7x + 3x2 + (9 + 12x + -21x2) = 0 Reorder the terms: -5 + 9 + 7x + 12x + 3x2 + -21x2 = 0 Combine like terms: -5 + 9 = 4 4 + 7x + 12x + 3x2 + -21x2 = 0 Combine like terms: 7x + 12x = 19x 4 + 19x + 3x2 + -21x2 = 0 Combine like terms: 3x2 + -21x2 = -18x2 4 + 19x + -18x2 = 0 Solving 4 + 19x + -18x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -18 the coefficient of the squared term: Divide each side by '-18'. -0.2222222222 + -1.055555556x + x2 = 0 Move the constant term to the right: Add '0.2222222222' to each side of the equation. -0.2222222222 + -1.055555556x + 0.2222222222 + x2 = 0 + 0.2222222222 Reorder the terms: -0.2222222222 + 0.2222222222 + -1.055555556x + x2 = 0 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + -1.055555556x + x2 = 0 + 0.2222222222 -1.055555556x + x2 = 0 + 0.2222222222 Combine like terms: 0 + 0.2222222222 = 0.2222222222 -1.055555556x + x2 = 0.2222222222 The x term is -1.055555556x. Take half its coefficient (-0.527777778). Square it (0.2785493830) and add it to both sides. Add '0.2785493830' to each side of the equation. -1.055555556x + 0.2785493830 + x2 = 0.2222222222 + 0.2785493830 Reorder the terms: 0.2785493830 + -1.055555556x + x2 = 0.2222222222 + 0.2785493830 Combine like terms: 0.2222222222 + 0.2785493830 = 0.5007716052 0.2785493830 + -1.055555556x + x2 = 0.5007716052 Factor a perfect square on the left side: (x + -0.527777778)(x + -0.527777778) = 0.5007716052 Calculate the square root of the right side: 0.707652178 Break this problem into two subproblems by setting (x + -0.527777778) equal to 0.707652178 and -0.707652178.Subproblem 1
x + -0.527777778 = 0.707652178 Simplifying x + -0.527777778 = 0.707652178 Reorder the terms: -0.527777778 + x = 0.707652178 Solving -0.527777778 + x = 0.707652178 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.527777778' to each side of the equation. -0.527777778 + 0.527777778 + x = 0.707652178 + 0.527777778 Combine like terms: -0.527777778 + 0.527777778 = 0.000000000 0.000000000 + x = 0.707652178 + 0.527777778 x = 0.707652178 + 0.527777778 Combine like terms: 0.707652178 + 0.527777778 = 1.235429956 x = 1.235429956 Simplifying x = 1.235429956Subproblem 2
x + -0.527777778 = -0.707652178 Simplifying x + -0.527777778 = -0.707652178 Reorder the terms: -0.527777778 + x = -0.707652178 Solving -0.527777778 + x = -0.707652178 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.527777778' to each side of the equation. -0.527777778 + 0.527777778 + x = -0.707652178 + 0.527777778 Combine like terms: -0.527777778 + 0.527777778 = 0.000000000 0.000000000 + x = -0.707652178 + 0.527777778 x = -0.707652178 + 0.527777778 Combine like terms: -0.707652178 + 0.527777778 = -0.1798744 x = -0.1798744 Simplifying x = -0.1798744Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.235429956, -0.1798744}
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