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Simplifying 9x2 + 12x + -3 = (3x + 1)(1x + -1) Reorder the terms: -3 + 12x + 9x2 = (3x + 1)(1x + -1) Reorder the terms: -3 + 12x + 9x2 = (1 + 3x)(1x + -1) Reorder the terms: -3 + 12x + 9x2 = (1 + 3x)(-1 + 1x) Multiply (1 + 3x) * (-1 + 1x) -3 + 12x + 9x2 = (1(-1 + 1x) + 3x * (-1 + 1x)) -3 + 12x + 9x2 = ((-1 * 1 + 1x * 1) + 3x * (-1 + 1x)) -3 + 12x + 9x2 = ((-1 + 1x) + 3x * (-1 + 1x)) -3 + 12x + 9x2 = (-1 + 1x + (-1 * 3x + 1x * 3x)) -3 + 12x + 9x2 = (-1 + 1x + (-3x + 3x2)) Combine like terms: 1x + -3x = -2x -3 + 12x + 9x2 = (-1 + -2x + 3x2) Solving -3 + 12x + 9x2 = -1 + -2x + 3x2 Solving for variable 'x'. Reorder the terms: -3 + 1 + 12x + 2x + 9x2 + -3x2 = -1 + -2x + 3x2 + 1 + 2x + -3x2 Combine like terms: -3 + 1 = -2 -2 + 12x + 2x + 9x2 + -3x2 = -1 + -2x + 3x2 + 1 + 2x + -3x2 Combine like terms: 12x + 2x = 14x -2 + 14x + 9x2 + -3x2 = -1 + -2x + 3x2 + 1 + 2x + -3x2 Combine like terms: 9x2 + -3x2 = 6x2 -2 + 14x + 6x2 = -1 + -2x + 3x2 + 1 + 2x + -3x2 Reorder the terms: -2 + 14x + 6x2 = -1 + 1 + -2x + 2x + 3x2 + -3x2 Combine like terms: -1 + 1 = 0 -2 + 14x + 6x2 = 0 + -2x + 2x + 3x2 + -3x2 -2 + 14x + 6x2 = -2x + 2x + 3x2 + -3x2 Combine like terms: -2x + 2x = 0 -2 + 14x + 6x2 = 0 + 3x2 + -3x2 -2 + 14x + 6x2 = 3x2 + -3x2 Combine like terms: 3x2 + -3x2 = 0 -2 + 14x + 6x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-1 + 7x + 3x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1 + 7x + 3x2)' equal to zero and attempt to solve: Simplifying -1 + 7x + 3x2 = 0 Solving -1 + 7x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + 2.333333333x + x2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + 2.333333333x + 0.3333333333 + x2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + 2.333333333x + x2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + 2.333333333x + x2 = 0 + 0.3333333333 2.333333333x + x2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 2.333333333x + x2 = 0.3333333333 The x term is 2.333333333x. Take half its coefficient (1.166666667). Square it (1.361111112) and add it to both sides. Add '1.361111112' to each side of the equation. 2.333333333x + 1.361111112 + x2 = 0.3333333333 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333x + x2 = 0.3333333333 + 1.361111112 Combine like terms: 0.3333333333 + 1.361111112 = 1.6944444453 1.361111112 + 2.333333333x + x2 = 1.6944444453 Factor a perfect square on the left side: (x + 1.166666667)(x + 1.166666667) = 1.6944444453 Calculate the square root of the right side: 1.30170828 Break this problem into two subproblems by setting (x + 1.166666667) equal to 1.30170828 and -1.30170828.Subproblem 1
x + 1.166666667 = 1.30170828 Simplifying x + 1.166666667 = 1.30170828 Reorder the terms: 1.166666667 + x = 1.30170828 Solving 1.166666667 + x = 1.30170828 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = 1.30170828 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = 1.30170828 + -1.166666667 x = 1.30170828 + -1.166666667 Combine like terms: 1.30170828 + -1.166666667 = 0.135041613 x = 0.135041613 Simplifying x = 0.135041613Subproblem 2
x + 1.166666667 = -1.30170828 Simplifying x + 1.166666667 = -1.30170828 Reorder the terms: 1.166666667 + x = -1.30170828 Solving 1.166666667 + x = -1.30170828 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.166666667' to each side of the equation. 1.166666667 + -1.166666667 + x = -1.30170828 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = -1.30170828 + -1.166666667 x = -1.30170828 + -1.166666667 Combine like terms: -1.30170828 + -1.166666667 = -2.468374947 x = -2.468374947 Simplifying x = -2.468374947Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.135041613, -2.468374947}Solution
x = {0.135041613, -2.468374947}
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