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Simplifying 4x2 + 3 = x2 + -7x Reorder the terms: 3 + 4x2 = x2 + -7x Reorder the terms: 3 + 4x2 = -7x + x2 Solving 3 + 4x2 = -7x + x2 Solving for variable 'x'. Reorder the terms: 3 + 7x + 4x2 + -1x2 = -7x + x2 + 7x + -1x2 Combine like terms: 4x2 + -1x2 = 3x2 3 + 7x + 3x2 = -7x + x2 + 7x + -1x2 Reorder the terms: 3 + 7x + 3x2 = -7x + 7x + x2 + -1x2 Combine like terms: -7x + 7x = 0 3 + 7x + 3x2 = 0 + x2 + -1x2 3 + 7x + 3x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 3 + 7x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1 + 2.333333333x + x2 = 0 Move the constant term to the right: Add '-1' to each side of the equation. 1 + 2.333333333x + -1 + x2 = 0 + -1 Reorder the terms: 1 + -1 + 2.333333333x + x2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 2.333333333x + x2 = 0 + -1 2.333333333x + x2 = 0 + -1 Combine like terms: 0 + -1 = -1 2.333333333x + x2 = -1 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 = -1 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333x + x2 = -1 + 1.361111112 Combine like terms: -1 + 1.361111112 = 0.361111112 1.361111112 + 2.333333333x + x2 = 0.361111112 Factor a perfect square on the left side: (x + 1.166666667)(x + 1.166666667) = 0.361111112 Calculate the square root of the right side: 0.600925213 Break this problem into two subproblems by setting (x + 1.166666667) equal to 0.600925213 and -0.600925213.Subproblem 1
x + 1.166666667 = 0.600925213 Simplifying x + 1.166666667 = 0.600925213 Reorder the terms: 1.166666667 + x = 0.600925213 Solving 1.166666667 + x = 0.600925213 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 = 0.600925213 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = 0.600925213 + -1.166666667 x = 0.600925213 + -1.166666667 Combine like terms: 0.600925213 + -1.166666667 = -0.565741454 x = -0.565741454 Simplifying x = -0.565741454Subproblem 2
x + 1.166666667 = -0.600925213 Simplifying x + 1.166666667 = -0.600925213 Reorder the terms: 1.166666667 + x = -0.600925213 Solving 1.166666667 + x = -0.600925213 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 = -0.600925213 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = -0.600925213 + -1.166666667 x = -0.600925213 + -1.166666667 Combine like terms: -0.600925213 + -1.166666667 = -1.76759188 x = -1.76759188 Simplifying x = -1.76759188Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.565741454, -1.76759188}
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