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Simplifying 7x2 + -14 + 6x2 + -5x = 180 Reorder the terms: -14 + -5x + 7x2 + 6x2 = 180 Combine like terms: 7x2 + 6x2 = 13x2 -14 + -5x + 13x2 = 180 Solving -14 + -5x + 13x2 = 180 Solving for variable 'x'. Reorder the terms: -14 + -180 + -5x + 13x2 = 180 + -180 Combine like terms: -14 + -180 = -194 -194 + -5x + 13x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -194 + -5x + 13x2 = 0 Begin completing the square. Divide all terms by 13 the coefficient of the squared term: Divide each side by '13'. -14.92307692 + -0.3846153846x + x2 = 0 Move the constant term to the right: Add '14.92307692' to each side of the equation. -14.92307692 + -0.3846153846x + 14.92307692 + x2 = 0 + 14.92307692 Reorder the terms: -14.92307692 + 14.92307692 + -0.3846153846x + x2 = 0 + 14.92307692 Combine like terms: -14.92307692 + 14.92307692 = 0.00000000 0.00000000 + -0.3846153846x + x2 = 0 + 14.92307692 -0.3846153846x + x2 = 0 + 14.92307692 Combine like terms: 0 + 14.92307692 = 14.92307692 -0.3846153846x + x2 = 14.92307692 The x term is -0.3846153846x. Take half its coefficient (-0.1923076923). Square it (0.03698224852) and add it to both sides. Add '0.03698224852' to each side of the equation. -0.3846153846x + 0.03698224852 + x2 = 14.92307692 + 0.03698224852 Reorder the terms: 0.03698224852 + -0.3846153846x + x2 = 14.92307692 + 0.03698224852 Combine like terms: 14.92307692 + 0.03698224852 = 14.96005916852 0.03698224852 + -0.3846153846x + x2 = 14.96005916852 Factor a perfect square on the left side: (x + -0.1923076923)(x + -0.1923076923) = 14.96005916852 Calculate the square root of the right side: 3.86782357 Break this problem into two subproblems by setting (x + -0.1923076923) equal to 3.86782357 and -3.86782357.Subproblem 1
x + -0.1923076923 = 3.86782357 Simplifying x + -0.1923076923 = 3.86782357 Reorder the terms: -0.1923076923 + x = 3.86782357 Solving -0.1923076923 + x = 3.86782357 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1923076923' to each side of the equation. -0.1923076923 + 0.1923076923 + x = 3.86782357 + 0.1923076923 Combine like terms: -0.1923076923 + 0.1923076923 = 0.0000000000 0.0000000000 + x = 3.86782357 + 0.1923076923 x = 3.86782357 + 0.1923076923 Combine like terms: 3.86782357 + 0.1923076923 = 4.0601312623 x = 4.0601312623 Simplifying x = 4.0601312623Subproblem 2
x + -0.1923076923 = -3.86782357 Simplifying x + -0.1923076923 = -3.86782357 Reorder the terms: -0.1923076923 + x = -3.86782357 Solving -0.1923076923 + x = -3.86782357 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1923076923' to each side of the equation. -0.1923076923 + 0.1923076923 + x = -3.86782357 + 0.1923076923 Combine like terms: -0.1923076923 + 0.1923076923 = 0.0000000000 0.0000000000 + x = -3.86782357 + 0.1923076923 x = -3.86782357 + 0.1923076923 Combine like terms: -3.86782357 + 0.1923076923 = -3.6755158777 x = -3.6755158777 Simplifying x = -3.6755158777Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.0601312623, -3.6755158777}
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