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Simplifying 2x(3x + -5) = 3(7 + -8x) Reorder the terms: 2x(-5 + 3x) = 3(7 + -8x) (-5 * 2x + 3x * 2x) = 3(7 + -8x) (-10x + 6x2) = 3(7 + -8x) -10x + 6x2 = (7 * 3 + -8x * 3) -10x + 6x2 = (21 + -24x) Solving -10x + 6x2 = 21 + -24x Solving for variable 'x'. Reorder the terms: -21 + -10x + 24x + 6x2 = 21 + -24x + -21 + 24x Combine like terms: -10x + 24x = 14x -21 + 14x + 6x2 = 21 + -24x + -21 + 24x Reorder the terms: -21 + 14x + 6x2 = 21 + -21 + -24x + 24x Combine like terms: 21 + -21 = 0 -21 + 14x + 6x2 = 0 + -24x + 24x -21 + 14x + 6x2 = -24x + 24x Combine like terms: -24x + 24x = 0 -21 + 14x + 6x2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -3.5 + 2.333333333x + x2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + 2.333333333x + 3.5 + x2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + 2.333333333x + x2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + 2.333333333x + x2 = 0 + 3.5 2.333333333x + x2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 2.333333333x + x2 = 3.5 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 = 3.5 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333x + x2 = 3.5 + 1.361111112 Combine like terms: 3.5 + 1.361111112 = 4.861111112 1.361111112 + 2.333333333x + x2 = 4.861111112 Factor a perfect square on the left side: (x + 1.166666667)(x + 1.166666667) = 4.861111112 Calculate the square root of the right side: 2.204792759 Break this problem into two subproblems by setting (x + 1.166666667) equal to 2.204792759 and -2.204792759.Subproblem 1
x + 1.166666667 = 2.204792759 Simplifying x + 1.166666667 = 2.204792759 Reorder the terms: 1.166666667 + x = 2.204792759 Solving 1.166666667 + x = 2.204792759 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 = 2.204792759 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = 2.204792759 + -1.166666667 x = 2.204792759 + -1.166666667 Combine like terms: 2.204792759 + -1.166666667 = 1.038126092 x = 1.038126092 Simplifying x = 1.038126092Subproblem 2
x + 1.166666667 = -2.204792759 Simplifying x + 1.166666667 = -2.204792759 Reorder the terms: 1.166666667 + x = -2.204792759 Solving 1.166666667 + x = -2.204792759 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 = -2.204792759 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = -2.204792759 + -1.166666667 x = -2.204792759 + -1.166666667 Combine like terms: -2.204792759 + -1.166666667 = -3.371459426 x = -3.371459426 Simplifying x = -3.371459426Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.038126092, -3.371459426}
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