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Simplifying (14x2 + 13x + 12) + -1(7x2 + -20x + 4) = 0 Reorder the terms: (12 + 13x + 14x2) + -1(7x2 + -20x + 4) = 0 Remove parenthesis around (12 + 13x + 14x2) 12 + 13x + 14x2 + -1(7x2 + -20x + 4) = 0 Reorder the terms: 12 + 13x + 14x2 + -1(4 + -20x + 7x2) = 0 12 + 13x + 14x2 + (4 * -1 + -20x * -1 + 7x2 * -1) = 0 12 + 13x + 14x2 + (-4 + 20x + -7x2) = 0 Reorder the terms: 12 + -4 + 13x + 20x + 14x2 + -7x2 = 0 Combine like terms: 12 + -4 = 8 8 + 13x + 20x + 14x2 + -7x2 = 0 Combine like terms: 13x + 20x = 33x 8 + 33x + 14x2 + -7x2 = 0 Combine like terms: 14x2 + -7x2 = 7x2 8 + 33x + 7x2 = 0 Solving 8 + 33x + 7x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 1.142857143 + 4.714285714x + x2 = 0 Move the constant term to the right: Add '-1.142857143' to each side of the equation. 1.142857143 + 4.714285714x + -1.142857143 + x2 = 0 + -1.142857143 Reorder the terms: 1.142857143 + -1.142857143 + 4.714285714x + x2 = 0 + -1.142857143 Combine like terms: 1.142857143 + -1.142857143 = 0.000000000 0.000000000 + 4.714285714x + x2 = 0 + -1.142857143 4.714285714x + x2 = 0 + -1.142857143 Combine like terms: 0 + -1.142857143 = -1.142857143 4.714285714x + x2 = -1.142857143 The x term is 4.714285714x. Take half its coefficient (2.357142857). Square it (5.556122448) and add it to both sides. Add '5.556122448' to each side of the equation. 4.714285714x + 5.556122448 + x2 = -1.142857143 + 5.556122448 Reorder the terms: 5.556122448 + 4.714285714x + x2 = -1.142857143 + 5.556122448 Combine like terms: -1.142857143 + 5.556122448 = 4.413265305 5.556122448 + 4.714285714x + x2 = 4.413265305 Factor a perfect square on the left side: (x + 2.357142857)(x + 2.357142857) = 4.413265305 Calculate the square root of the right side: 2.10077731 Break this problem into two subproblems by setting (x + 2.357142857) equal to 2.10077731 and -2.10077731.Subproblem 1
x + 2.357142857 = 2.10077731 Simplifying x + 2.357142857 = 2.10077731 Reorder the terms: 2.357142857 + x = 2.10077731 Solving 2.357142857 + x = 2.10077731 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.357142857' to each side of the equation. 2.357142857 + -2.357142857 + x = 2.10077731 + -2.357142857 Combine like terms: 2.357142857 + -2.357142857 = 0.000000000 0.000000000 + x = 2.10077731 + -2.357142857 x = 2.10077731 + -2.357142857 Combine like terms: 2.10077731 + -2.357142857 = -0.256365547 x = -0.256365547 Simplifying x = -0.256365547Subproblem 2
x + 2.357142857 = -2.10077731 Simplifying x + 2.357142857 = -2.10077731 Reorder the terms: 2.357142857 + x = -2.10077731 Solving 2.357142857 + x = -2.10077731 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.357142857' to each side of the equation. 2.357142857 + -2.357142857 + x = -2.10077731 + -2.357142857 Combine like terms: 2.357142857 + -2.357142857 = 0.000000000 0.000000000 + x = -2.10077731 + -2.357142857 x = -2.10077731 + -2.357142857 Combine like terms: -2.10077731 + -2.357142857 = -4.457920167 x = -4.457920167 Simplifying x = -4.457920167Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.256365547, -4.457920167}
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