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Simplifying 7(3x2 + 7x + 4) + -2(5x2 + 5x + 3) = 0 Reorder the terms: 7(4 + 7x + 3x2) + -2(5x2 + 5x + 3) = 0 (4 * 7 + 7x * 7 + 3x2 * 7) + -2(5x2 + 5x + 3) = 0 (28 + 49x + 21x2) + -2(5x2 + 5x + 3) = 0 Reorder the terms: 28 + 49x + 21x2 + -2(3 + 5x + 5x2) = 0 28 + 49x + 21x2 + (3 * -2 + 5x * -2 + 5x2 * -2) = 0 28 + 49x + 21x2 + (-6 + -10x + -10x2) = 0 Reorder the terms: 28 + -6 + 49x + -10x + 21x2 + -10x2 = 0 Combine like terms: 28 + -6 = 22 22 + 49x + -10x + 21x2 + -10x2 = 0 Combine like terms: 49x + -10x = 39x 22 + 39x + 21x2 + -10x2 = 0 Combine like terms: 21x2 + -10x2 = 11x2 22 + 39x + 11x2 = 0 Solving 22 + 39x + 11x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. 2 + 3.545454545x + x2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + 3.545454545x + -2 + x2 = 0 + -2 Reorder the terms: 2 + -2 + 3.545454545x + x2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + 3.545454545x + x2 = 0 + -2 3.545454545x + x2 = 0 + -2 Combine like terms: 0 + -2 = -2 3.545454545x + x2 = -2 The x term is 3.545454545x. Take half its coefficient (1.772727273). Square it (3.142561984) and add it to both sides. Add '3.142561984' to each side of the equation. 3.545454545x + 3.142561984 + x2 = -2 + 3.142561984 Reorder the terms: 3.142561984 + 3.545454545x + x2 = -2 + 3.142561984 Combine like terms: -2 + 3.142561984 = 1.142561984 3.142561984 + 3.545454545x + x2 = 1.142561984 Factor a perfect square on the left side: (x + 1.772727273)(x + 1.772727273) = 1.142561984 Calculate the square root of the right side: 1.068906911 Break this problem into two subproblems by setting (x + 1.772727273) equal to 1.068906911 and -1.068906911.Subproblem 1
x + 1.772727273 = 1.068906911 Simplifying x + 1.772727273 = 1.068906911 Reorder the terms: 1.772727273 + x = 1.068906911 Solving 1.772727273 + x = 1.068906911 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.772727273' to each side of the equation. 1.772727273 + -1.772727273 + x = 1.068906911 + -1.772727273 Combine like terms: 1.772727273 + -1.772727273 = 0.000000000 0.000000000 + x = 1.068906911 + -1.772727273 x = 1.068906911 + -1.772727273 Combine like terms: 1.068906911 + -1.772727273 = -0.703820362 x = -0.703820362 Simplifying x = -0.703820362Subproblem 2
x + 1.772727273 = -1.068906911 Simplifying x + 1.772727273 = -1.068906911 Reorder the terms: 1.772727273 + x = -1.068906911 Solving 1.772727273 + x = -1.068906911 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.772727273' to each side of the equation. 1.772727273 + -1.772727273 + x = -1.068906911 + -1.772727273 Combine like terms: 1.772727273 + -1.772727273 = 0.000000000 0.000000000 + x = -1.068906911 + -1.772727273 x = -1.068906911 + -1.772727273 Combine like terms: -1.068906911 + -1.772727273 = -2.841634184 x = -2.841634184 Simplifying x = -2.841634184Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.703820362, -2.841634184}
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