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Simplifying 3x(2x + -5) + 4(3x + -7) = 0 Reorder the terms: 3x(-5 + 2x) + 4(3x + -7) = 0 (-5 * 3x + 2x * 3x) + 4(3x + -7) = 0 (-15x + 6x2) + 4(3x + -7) = 0 Reorder the terms: -15x + 6x2 + 4(-7 + 3x) = 0 -15x + 6x2 + (-7 * 4 + 3x * 4) = 0 -15x + 6x2 + (-28 + 12x) = 0 Reorder the terms: -28 + -15x + 12x + 6x2 = 0 Combine like terms: -15x + 12x = -3x -28 + -3x + 6x2 = 0 Solving -28 + -3x + 6x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -4.666666667 + -0.5x + x2 = 0 Move the constant term to the right: Add '4.666666667' to each side of the equation. -4.666666667 + -0.5x + 4.666666667 + x2 = 0 + 4.666666667 Reorder the terms: -4.666666667 + 4.666666667 + -0.5x + x2 = 0 + 4.666666667 Combine like terms: -4.666666667 + 4.666666667 = 0.000000000 0.000000000 + -0.5x + x2 = 0 + 4.666666667 -0.5x + x2 = 0 + 4.666666667 Combine like terms: 0 + 4.666666667 = 4.666666667 -0.5x + x2 = 4.666666667 The x term is -0.5x. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5x + 0.0625 + x2 = 4.666666667 + 0.0625 Reorder the terms: 0.0625 + -0.5x + x2 = 4.666666667 + 0.0625 Combine like terms: 4.666666667 + 0.0625 = 4.729166667 0.0625 + -0.5x + x2 = 4.729166667 Factor a perfect square on the left side: (x + -0.25)(x + -0.25) = 4.729166667 Calculate the square root of the right side: 2.174664725 Break this problem into two subproblems by setting (x + -0.25) equal to 2.174664725 and -2.174664725.Subproblem 1
x + -0.25 = 2.174664725 Simplifying x + -0.25 = 2.174664725 Reorder the terms: -0.25 + x = 2.174664725 Solving -0.25 + x = 2.174664725 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = 2.174664725 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 2.174664725 + 0.25 x = 2.174664725 + 0.25 Combine like terms: 2.174664725 + 0.25 = 2.424664725 x = 2.424664725 Simplifying x = 2.424664725Subproblem 2
x + -0.25 = -2.174664725 Simplifying x + -0.25 = -2.174664725 Reorder the terms: -0.25 + x = -2.174664725 Solving -0.25 + x = -2.174664725 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + x = -2.174664725 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -2.174664725 + 0.25 x = -2.174664725 + 0.25 Combine like terms: -2.174664725 + 0.25 = -1.924664725 x = -1.924664725 Simplifying x = -1.924664725Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.424664725, -1.924664725}
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