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Simplifying 3x2 + 20x + 7 = -4x2 + -4x + 24 Reorder the terms: 7 + 20x + 3x2 = -4x2 + -4x + 24 Reorder the terms: 7 + 20x + 3x2 = 24 + -4x + -4x2 Solving 7 + 20x + 3x2 = 24 + -4x + -4x2 Solving for variable 'x'. Reorder the terms: 7 + -24 + 20x + 4x + 3x2 + 4x2 = 24 + -4x + -4x2 + -24 + 4x + 4x2 Combine like terms: 7 + -24 = -17 -17 + 20x + 4x + 3x2 + 4x2 = 24 + -4x + -4x2 + -24 + 4x + 4x2 Combine like terms: 20x + 4x = 24x -17 + 24x + 3x2 + 4x2 = 24 + -4x + -4x2 + -24 + 4x + 4x2 Combine like terms: 3x2 + 4x2 = 7x2 -17 + 24x + 7x2 = 24 + -4x + -4x2 + -24 + 4x + 4x2 Reorder the terms: -17 + 24x + 7x2 = 24 + -24 + -4x + 4x + -4x2 + 4x2 Combine like terms: 24 + -24 = 0 -17 + 24x + 7x2 = 0 + -4x + 4x + -4x2 + 4x2 -17 + 24x + 7x2 = -4x + 4x + -4x2 + 4x2 Combine like terms: -4x + 4x = 0 -17 + 24x + 7x2 = 0 + -4x2 + 4x2 -17 + 24x + 7x2 = -4x2 + 4x2 Combine like terms: -4x2 + 4x2 = 0 -17 + 24x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -2.428571429 + 3.428571429x + x2 = 0 Move the constant term to the right: Add '2.428571429' to each side of the equation. -2.428571429 + 3.428571429x + 2.428571429 + x2 = 0 + 2.428571429 Reorder the terms: -2.428571429 + 2.428571429 + 3.428571429x + x2 = 0 + 2.428571429 Combine like terms: -2.428571429 + 2.428571429 = 0.000000000 0.000000000 + 3.428571429x + x2 = 0 + 2.428571429 3.428571429x + x2 = 0 + 2.428571429 Combine like terms: 0 + 2.428571429 = 2.428571429 3.428571429x + x2 = 2.428571429 The x term is 3.428571429x. Take half its coefficient (1.714285715). Square it (2.938775513) and add it to both sides. Add '2.938775513' to each side of the equation. 3.428571429x + 2.938775513 + x2 = 2.428571429 + 2.938775513 Reorder the terms: 2.938775513 + 3.428571429x + x2 = 2.428571429 + 2.938775513 Combine like terms: 2.428571429 + 2.938775513 = 5.367346942 2.938775513 + 3.428571429x + x2 = 5.367346942 Factor a perfect square on the left side: (x + 1.714285715)(x + 1.714285715) = 5.367346942 Calculate the square root of the right side: 2.316753535 Break this problem into two subproblems by setting (x + 1.714285715) equal to 2.316753535 and -2.316753535.Subproblem 1
x + 1.714285715 = 2.316753535 Simplifying x + 1.714285715 = 2.316753535 Reorder the terms: 1.714285715 + x = 2.316753535 Solving 1.714285715 + x = 2.316753535 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.714285715' to each side of the equation. 1.714285715 + -1.714285715 + x = 2.316753535 + -1.714285715 Combine like terms: 1.714285715 + -1.714285715 = 0.000000000 0.000000000 + x = 2.316753535 + -1.714285715 x = 2.316753535 + -1.714285715 Combine like terms: 2.316753535 + -1.714285715 = 0.60246782 x = 0.60246782 Simplifying x = 0.60246782Subproblem 2
x + 1.714285715 = -2.316753535 Simplifying x + 1.714285715 = -2.316753535 Reorder the terms: 1.714285715 + x = -2.316753535 Solving 1.714285715 + x = -2.316753535 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.714285715' to each side of the equation. 1.714285715 + -1.714285715 + x = -2.316753535 + -1.714285715 Combine like terms: 1.714285715 + -1.714285715 = 0.000000000 0.000000000 + x = -2.316753535 + -1.714285715 x = -2.316753535 + -1.714285715 Combine like terms: -2.316753535 + -1.714285715 = -4.03103925 x = -4.03103925 Simplifying x = -4.03103925Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.60246782, -4.03103925}
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