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Simplifying 7(2x2 + 7x + 3) + -7(4x2 + 2x + 2) = 0 Reorder the terms: 7(3 + 7x + 2x2) + -7(4x2 + 2x + 2) = 0 (3 * 7 + 7x * 7 + 2x2 * 7) + -7(4x2 + 2x + 2) = 0 (21 + 49x + 14x2) + -7(4x2 + 2x + 2) = 0 Reorder the terms: 21 + 49x + 14x2 + -7(2 + 2x + 4x2) = 0 21 + 49x + 14x2 + (2 * -7 + 2x * -7 + 4x2 * -7) = 0 21 + 49x + 14x2 + (-14 + -14x + -28x2) = 0 Reorder the terms: 21 + -14 + 49x + -14x + 14x2 + -28x2 = 0 Combine like terms: 21 + -14 = 7 7 + 49x + -14x + 14x2 + -28x2 = 0 Combine like terms: 49x + -14x = 35x 7 + 35x + 14x2 + -28x2 = 0 Combine like terms: 14x2 + -28x2 = -14x2 7 + 35x + -14x2 = 0 Solving 7 + 35x + -14x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '7'. 7(1 + 5x + -2x2) = 0 Ignore the factor 7.Subproblem 1
Set the factor '(1 + 5x + -2x2)' equal to zero and attempt to solve: Simplifying 1 + 5x + -2x2 = 0 Solving 1 + 5x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -0.5 + -2.5x + x2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + -2.5x + 0.5 + x2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + -2.5x + x2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + -2.5x + x2 = 0 + 0.5 -2.5x + x2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 -2.5x + x2 = 0.5 The x term is -2.5x. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5x + 1.5625 + x2 = 0.5 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = 0.5 + 1.5625 Combine like terms: 0.5 + 1.5625 = 2.0625 1.5625 + -2.5x + x2 = 2.0625 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 2.0625 Calculate the square root of the right side: 1.436140662 Break this problem into two subproblems by setting (x + -1.25) equal to 1.436140662 and -1.436140662.Subproblem 1
x + -1.25 = 1.436140662 Simplifying x + -1.25 = 1.436140662 Reorder the terms: -1.25 + x = 1.436140662 Solving -1.25 + x = 1.436140662 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 1.436140662 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 1.436140662 + 1.25 x = 1.436140662 + 1.25 Combine like terms: 1.436140662 + 1.25 = 2.686140662 x = 2.686140662 Simplifying x = 2.686140662Subproblem 2
x + -1.25 = -1.436140662 Simplifying x + -1.25 = -1.436140662 Reorder the terms: -1.25 + x = -1.436140662 Solving -1.25 + x = -1.436140662 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -1.436140662 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -1.436140662 + 1.25 x = -1.436140662 + 1.25 Combine like terms: -1.436140662 + 1.25 = -0.186140662 x = -0.186140662 Simplifying x = -0.186140662Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.686140662, -0.186140662}Solution
x = {2.686140662, -0.186140662}
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