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Simplifying (7x2 + -5x + -6) * 6x2 = 0 Reorder the terms: (-6 + -5x + 7x2) * 6x2 = 0 Reorder the terms for easier multiplication: 6x2(-6 + -5x + 7x2) = 0 (-6 * 6x2 + -5x * 6x2 + 7x2 * 6x2) = 0 (-36x2 + -30x3 + 42x4) = 0 Solving -36x2 + -30x3 + 42x4 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '6x2'. 6x2(-6 + -5x + 7x2) = 0 Ignore the factor 6.Subproblem 1
Set the factor 'x2' equal to zero and attempt to solve: Simplifying x2 = 0 Solving x2 = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x2 = 0 Take the square root of each side: x = {0}Subproblem 2
Set the factor '(-6 + -5x + 7x2)' equal to zero and attempt to solve: Simplifying -6 + -5x + 7x2 = 0 Solving -6 + -5x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.8571428571 + -0.7142857143x + x2 = 0 Move the constant term to the right: Add '0.8571428571' to each side of the equation. -0.8571428571 + -0.7142857143x + 0.8571428571 + x2 = 0 + 0.8571428571 Reorder the terms: -0.8571428571 + 0.8571428571 + -0.7142857143x + x2 = 0 + 0.8571428571 Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000 0.0000000000 + -0.7142857143x + x2 = 0 + 0.8571428571 -0.7142857143x + x2 = 0 + 0.8571428571 Combine like terms: 0 + 0.8571428571 = 0.8571428571 -0.7142857143x + x2 = 0.8571428571 The x term is -0.7142857143x. Take half its coefficient (-0.3571428572). Square it (0.1275510204) and add it to both sides. Add '0.1275510204' to each side of the equation. -0.7142857143x + 0.1275510204 + x2 = 0.8571428571 + 0.1275510204 Reorder the terms: 0.1275510204 + -0.7142857143x + x2 = 0.8571428571 + 0.1275510204 Combine like terms: 0.8571428571 + 0.1275510204 = 0.9846938775 0.1275510204 + -0.7142857143x + x2 = 0.9846938775 Factor a perfect square on the left side: (x + -0.3571428572)(x + -0.3571428572) = 0.9846938775 Calculate the square root of the right side: 0.992317428 Break this problem into two subproblems by setting (x + -0.3571428572) equal to 0.992317428 and -0.992317428.Subproblem 1
x + -0.3571428572 = 0.992317428 Simplifying x + -0.3571428572 = 0.992317428 Reorder the terms: -0.3571428572 + x = 0.992317428 Solving -0.3571428572 + x = 0.992317428 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + x = 0.992317428 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + x = 0.992317428 + 0.3571428572 x = 0.992317428 + 0.3571428572 Combine like terms: 0.992317428 + 0.3571428572 = 1.3494602852 x = 1.3494602852 Simplifying x = 1.3494602852Subproblem 2
x + -0.3571428572 = -0.992317428 Simplifying x + -0.3571428572 = -0.992317428 Reorder the terms: -0.3571428572 + x = -0.992317428 Solving -0.3571428572 + x = -0.992317428 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3571428572' to each side of the equation. -0.3571428572 + 0.3571428572 + x = -0.992317428 + 0.3571428572 Combine like terms: -0.3571428572 + 0.3571428572 = 0.0000000000 0.0000000000 + x = -0.992317428 + 0.3571428572 x = -0.992317428 + 0.3571428572 Combine like terms: -0.992317428 + 0.3571428572 = -0.6351745708 x = -0.6351745708 Simplifying x = -0.6351745708Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.3494602852, -0.6351745708}Solution
x = {0, 1.3494602852, -0.6351745708}
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