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Simplifying 3 + 2x(3x + -4) = 2x + 12 + -1(x + -7) Reorder the terms: 3 + 2x(-4 + 3x) = 2x + 12 + -1(x + -7) 3 + (-4 * 2x + 3x * 2x) = 2x + 12 + -1(x + -7) 3 + (-8x + 6x2) = 2x + 12 + -1(x + -7) Reorder the terms: 3 + -8x + 6x2 = 2x + 12 + -1(-7 + x) 3 + -8x + 6x2 = 2x + 12 + (-7 * -1 + x * -1) 3 + -8x + 6x2 = 2x + 12 + (7 + -1x) Reorder the terms: 3 + -8x + 6x2 = 12 + 7 + 2x + -1x Combine like terms: 12 + 7 = 19 3 + -8x + 6x2 = 19 + 2x + -1x Combine like terms: 2x + -1x = 1x 3 + -8x + 6x2 = 19 + 1x Solving 3 + -8x + 6x2 = 19 + 1x Solving for variable 'x'. Reorder the terms: 3 + -19 + -8x + -1x + 6x2 = 19 + 1x + -19 + -1x Combine like terms: 3 + -19 = -16 -16 + -8x + -1x + 6x2 = 19 + 1x + -19 + -1x Combine like terms: -8x + -1x = -9x -16 + -9x + 6x2 = 19 + 1x + -19 + -1x Reorder the terms: -16 + -9x + 6x2 = 19 + -19 + 1x + -1x Combine like terms: 19 + -19 = 0 -16 + -9x + 6x2 = 0 + 1x + -1x -16 + -9x + 6x2 = 1x + -1x Combine like terms: 1x + -1x = 0 -16 + -9x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2.666666667 + -1.5x + x2 = 0 Move the constant term to the right: Add '2.666666667' to each side of the equation. -2.666666667 + -1.5x + 2.666666667 + x2 = 0 + 2.666666667 Reorder the terms: -2.666666667 + 2.666666667 + -1.5x + x2 = 0 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + -1.5x + x2 = 0 + 2.666666667 -1.5x + x2 = 0 + 2.666666667 Combine like terms: 0 + 2.666666667 = 2.666666667 -1.5x + x2 = 2.666666667 The x term is -1.5x. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5x + 0.5625 + x2 = 2.666666667 + 0.5625 Reorder the terms: 0.5625 + -1.5x + x2 = 2.666666667 + 0.5625 Combine like terms: 2.666666667 + 0.5625 = 3.229166667 0.5625 + -1.5x + x2 = 3.229166667 Factor a perfect square on the left side: (x + -0.75)(x + -0.75) = 3.229166667 Calculate the square root of the right side: 1.796988221 Break this problem into two subproblems by setting (x + -0.75) equal to 1.796988221 and -1.796988221.Subproblem 1
x + -0.75 = 1.796988221 Simplifying x + -0.75 = 1.796988221 Reorder the terms: -0.75 + x = 1.796988221 Solving -0.75 + x = 1.796988221 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = 1.796988221 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = 1.796988221 + 0.75 x = 1.796988221 + 0.75 Combine like terms: 1.796988221 + 0.75 = 2.546988221 x = 2.546988221 Simplifying x = 2.546988221Subproblem 2
x + -0.75 = -1.796988221 Simplifying x + -0.75 = -1.796988221 Reorder the terms: -0.75 + x = -1.796988221 Solving -0.75 + x = -1.796988221 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + x = -1.796988221 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + x = -1.796988221 + 0.75 x = -1.796988221 + 0.75 Combine like terms: -1.796988221 + 0.75 = -1.046988221 x = -1.046988221 Simplifying x = -1.046988221Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.546988221, -1.046988221}
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