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Simplifying 8x(3x + 7) = 23 Reorder the terms: 8x(7 + 3x) = 23 (7 * 8x + 3x * 8x) = 23 (56x + 24x2) = 23 Solving 56x + 24x2 = 23 Solving for variable 'x'. Reorder the terms: -23 + 56x + 24x2 = 23 + -23 Combine like terms: 23 + -23 = 0 -23 + 56x + 24x2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -0.9583333333 + 2.333333333x + x2 = 0 Move the constant term to the right: Add '0.9583333333' to each side of the equation. -0.9583333333 + 2.333333333x + 0.9583333333 + x2 = 0 + 0.9583333333 Reorder the terms: -0.9583333333 + 0.9583333333 + 2.333333333x + x2 = 0 + 0.9583333333 Combine like terms: -0.9583333333 + 0.9583333333 = 0.0000000000 0.0000000000 + 2.333333333x + x2 = 0 + 0.9583333333 2.333333333x + x2 = 0 + 0.9583333333 Combine like terms: 0 + 0.9583333333 = 0.9583333333 2.333333333x + x2 = 0.9583333333 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 = 0.9583333333 + 1.361111112 Reorder the terms: 1.361111112 + 2.333333333x + x2 = 0.9583333333 + 1.361111112 Combine like terms: 0.9583333333 + 1.361111112 = 2.3194444453 1.361111112 + 2.333333333x + x2 = 2.3194444453 Factor a perfect square on the left side: (x + 1.166666667)(x + 1.166666667) = 2.3194444453 Calculate the square root of the right side: 1.52297224 Break this problem into two subproblems by setting (x + 1.166666667) equal to 1.52297224 and -1.52297224.Subproblem 1
x + 1.166666667 = 1.52297224 Simplifying x + 1.166666667 = 1.52297224 Reorder the terms: 1.166666667 + x = 1.52297224 Solving 1.166666667 + x = 1.52297224 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 = 1.52297224 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = 1.52297224 + -1.166666667 x = 1.52297224 + -1.166666667 Combine like terms: 1.52297224 + -1.166666667 = 0.356305573 x = 0.356305573 Simplifying x = 0.356305573Subproblem 2
x + 1.166666667 = -1.52297224 Simplifying x + 1.166666667 = -1.52297224 Reorder the terms: 1.166666667 + x = -1.52297224 Solving 1.166666667 + x = -1.52297224 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 = -1.52297224 + -1.166666667 Combine like terms: 1.166666667 + -1.166666667 = 0.000000000 0.000000000 + x = -1.52297224 + -1.166666667 x = -1.52297224 + -1.166666667 Combine like terms: -1.52297224 + -1.166666667 = -2.689638907 x = -2.689638907 Simplifying x = -2.689638907Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.356305573, -2.689638907}
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