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Simplifying 4(x + 2)(3x + -7) = 22x + -65 Reorder the terms: 4(2 + x)(3x + -7) = 22x + -65 Reorder the terms: 4(2 + x)(-7 + 3x) = 22x + -65 Multiply (2 + x) * (-7 + 3x) 4(2(-7 + 3x) + x(-7 + 3x)) = 22x + -65 4((-7 * 2 + 3x * 2) + x(-7 + 3x)) = 22x + -65 4((-14 + 6x) + x(-7 + 3x)) = 22x + -65 4(-14 + 6x + (-7 * x + 3x * x)) = 22x + -65 4(-14 + 6x + (-7x + 3x2)) = 22x + -65 Combine like terms: 6x + -7x = -1x 4(-14 + -1x + 3x2) = 22x + -65 (-14 * 4 + -1x * 4 + 3x2 * 4) = 22x + -65 (-56 + -4x + 12x2) = 22x + -65 Reorder the terms: -56 + -4x + 12x2 = -65 + 22x Solving -56 + -4x + 12x2 = -65 + 22x Solving for variable 'x'. Reorder the terms: -56 + 65 + -4x + -22x + 12x2 = -65 + 22x + 65 + -22x Combine like terms: -56 + 65 = 9 9 + -4x + -22x + 12x2 = -65 + 22x + 65 + -22x Combine like terms: -4x + -22x = -26x 9 + -26x + 12x2 = -65 + 22x + 65 + -22x Reorder the terms: 9 + -26x + 12x2 = -65 + 65 + 22x + -22x Combine like terms: -65 + 65 = 0 9 + -26x + 12x2 = 0 + 22x + -22x 9 + -26x + 12x2 = 22x + -22x Combine like terms: 22x + -22x = 0 9 + -26x + 12x2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 0.75 + -2.166666667x + x2 = 0 Move the constant term to the right: Add '-0.75' to each side of the equation. 0.75 + -2.166666667x + -0.75 + x2 = 0 + -0.75 Reorder the terms: 0.75 + -0.75 + -2.166666667x + x2 = 0 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + -2.166666667x + x2 = 0 + -0.75 -2.166666667x + x2 = 0 + -0.75 Combine like terms: 0 + -0.75 = -0.75 -2.166666667x + x2 = -0.75 The x term is -2.166666667x. Take half its coefficient (-1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. -2.166666667x + 1.173611113 + x2 = -0.75 + 1.173611113 Reorder the terms: 1.173611113 + -2.166666667x + x2 = -0.75 + 1.173611113 Combine like terms: -0.75 + 1.173611113 = 0.423611113 1.173611113 + -2.166666667x + x2 = 0.423611113 Factor a perfect square on the left side: (x + -1.083333334)(x + -1.083333334) = 0.423611113 Calculate the square root of the right side: 0.650854141 Break this problem into two subproblems by setting (x + -1.083333334) equal to 0.650854141 and -0.650854141.Subproblem 1
x + -1.083333334 = 0.650854141 Simplifying x + -1.083333334 = 0.650854141 Reorder the terms: -1.083333334 + x = 0.650854141 Solving -1.083333334 + x = 0.650854141 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + x = 0.650854141 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + x = 0.650854141 + 1.083333334 x = 0.650854141 + 1.083333334 Combine like terms: 0.650854141 + 1.083333334 = 1.734187475 x = 1.734187475 Simplifying x = 1.734187475Subproblem 2
x + -1.083333334 = -0.650854141 Simplifying x + -1.083333334 = -0.650854141 Reorder the terms: -1.083333334 + x = -0.650854141 Solving -1.083333334 + x = -0.650854141 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.083333334' to each side of the equation. -1.083333334 + 1.083333334 + x = -0.650854141 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + x = -0.650854141 + 1.083333334 x = -0.650854141 + 1.083333334 Combine like terms: -0.650854141 + 1.083333334 = 0.432479193 x = 0.432479193 Simplifying x = 0.432479193Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.734187475, 0.432479193}
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