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Simplifying 13 = 3 + (5x + -2)(-2x + 5) Reorder the terms: 13 = 3 + (-2 + 5x)(-2x + 5) Reorder the terms: 13 = 3 + (-2 + 5x)(5 + -2x) Multiply (-2 + 5x) * (5 + -2x) 13 = 3 + (-2(5 + -2x) + 5x * (5 + -2x)) 13 = 3 + ((5 * -2 + -2x * -2) + 5x * (5 + -2x)) 13 = 3 + ((-10 + 4x) + 5x * (5 + -2x)) 13 = 3 + (-10 + 4x + (5 * 5x + -2x * 5x)) 13 = 3 + (-10 + 4x + (25x + -10x2)) Combine like terms: 4x + 25x = 29x 13 = 3 + (-10 + 29x + -10x2) Combine like terms: 3 + -10 = -7 13 = -7 + 29x + -10x2 Solving 13 = -7 + 29x + -10x2 Solving for variable 'x'. Combine like terms: 13 + 7 = 20 20 + -29x + 10x2 = -7 + 29x + -10x2 + 7 + -29x + 10x2 Reorder the terms: 20 + -29x + 10x2 = -7 + 7 + 29x + -29x + -10x2 + 10x2 Combine like terms: -7 + 7 = 0 20 + -29x + 10x2 = 0 + 29x + -29x + -10x2 + 10x2 20 + -29x + 10x2 = 29x + -29x + -10x2 + 10x2 Combine like terms: 29x + -29x = 0 20 + -29x + 10x2 = 0 + -10x2 + 10x2 20 + -29x + 10x2 = -10x2 + 10x2 Combine like terms: -10x2 + 10x2 = 0 20 + -29x + 10x2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 2 + -2.9x + x2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + -2.9x + -2 + x2 = 0 + -2 Reorder the terms: 2 + -2 + -2.9x + x2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -2.9x + x2 = 0 + -2 -2.9x + x2 = 0 + -2 Combine like terms: 0 + -2 = -2 -2.9x + x2 = -2 The x term is -2.9x. Take half its coefficient (-1.45). Square it (2.1025) and add it to both sides. Add '2.1025' to each side of the equation. -2.9x + 2.1025 + x2 = -2 + 2.1025 Reorder the terms: 2.1025 + -2.9x + x2 = -2 + 2.1025 Combine like terms: -2 + 2.1025 = 0.1025 2.1025 + -2.9x + x2 = 0.1025 Factor a perfect square on the left side: (x + -1.45)(x + -1.45) = 0.1025 Calculate the square root of the right side: 0.320156212 Break this problem into two subproblems by setting (x + -1.45) equal to 0.320156212 and -0.320156212.Subproblem 1
x + -1.45 = 0.320156212 Simplifying x + -1.45 = 0.320156212 Reorder the terms: -1.45 + x = 0.320156212 Solving -1.45 + x = 0.320156212 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.45' to each side of the equation. -1.45 + 1.45 + x = 0.320156212 + 1.45 Combine like terms: -1.45 + 1.45 = 0.00 0.00 + x = 0.320156212 + 1.45 x = 0.320156212 + 1.45 Combine like terms: 0.320156212 + 1.45 = 1.770156212 x = 1.770156212 Simplifying x = 1.770156212Subproblem 2
x + -1.45 = -0.320156212 Simplifying x + -1.45 = -0.320156212 Reorder the terms: -1.45 + x = -0.320156212 Solving -1.45 + x = -0.320156212 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.45' to each side of the equation. -1.45 + 1.45 + x = -0.320156212 + 1.45 Combine like terms: -1.45 + 1.45 = 0.00 0.00 + x = -0.320156212 + 1.45 x = -0.320156212 + 1.45 Combine like terms: -0.320156212 + 1.45 = 1.129843788 x = 1.129843788 Simplifying x = 1.129843788Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.770156212, 1.129843788}
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