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Simplifying 420 = 16x * 16x + 27x Reorder the terms for easier multiplication: 420 = 16 * 16x * x + 27x Multiply 16 * 16 420 = 256x * x + 27x Multiply x * x 420 = 256x2 + 27x Reorder the terms: 420 = 27x + 256x2 Solving 420 = 27x + 256x2 Solving for variable 'x'. Reorder the terms: 420 + -27x + -256x2 = 27x + -27x + 256x2 + -256x2 Combine like terms: 27x + -27x = 0 420 + -27x + -256x2 = 0 + 256x2 + -256x2 420 + -27x + -256x2 = 256x2 + -256x2 Combine like terms: 256x2 + -256x2 = 0 420 + -27x + -256x2 = 0 Begin completing the square. Divide all terms by -256 the coefficient of the squared term: Divide each side by '-256'. -1.640625 + 0.10546875x + x2 = 0 Move the constant term to the right: Add '1.640625' to each side of the equation. -1.640625 + 0.10546875x + 1.640625 + x2 = 0 + 1.640625 Reorder the terms: -1.640625 + 1.640625 + 0.10546875x + x2 = 0 + 1.640625 Combine like terms: -1.640625 + 1.640625 = 0.000000 0.000000 + 0.10546875x + x2 = 0 + 1.640625 0.10546875x + x2 = 0 + 1.640625 Combine like terms: 0 + 1.640625 = 1.640625 0.10546875x + x2 = 1.640625 The x term is 0.10546875x. Take half its coefficient (0.052734375). Square it (0.002780914307) and add it to both sides. Add '0.002780914307' to each side of the equation. 0.10546875x + 0.002780914307 + x2 = 1.640625 + 0.002780914307 Reorder the terms: 0.002780914307 + 0.10546875x + x2 = 1.640625 + 0.002780914307 Combine like terms: 1.640625 + 0.002780914307 = 1.643405914307 0.002780914307 + 0.10546875x + x2 = 1.643405914307 Factor a perfect square on the left side: (x + 0.052734375)(x + 0.052734375) = 1.643405914307 Calculate the square root of the right side: 1.281953944 Break this problem into two subproblems by setting (x + 0.052734375) equal to 1.281953944 and -1.281953944.Subproblem 1
x + 0.052734375 = 1.281953944 Simplifying x + 0.052734375 = 1.281953944 Reorder the terms: 0.052734375 + x = 1.281953944 Solving 0.052734375 + x = 1.281953944 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.052734375' to each side of the equation. 0.052734375 + -0.052734375 + x = 1.281953944 + -0.052734375 Combine like terms: 0.052734375 + -0.052734375 = 0.000000000 0.000000000 + x = 1.281953944 + -0.052734375 x = 1.281953944 + -0.052734375 Combine like terms: 1.281953944 + -0.052734375 = 1.229219569 x = 1.229219569 Simplifying x = 1.229219569Subproblem 2
x + 0.052734375 = -1.281953944 Simplifying x + 0.052734375 = -1.281953944 Reorder the terms: 0.052734375 + x = -1.281953944 Solving 0.052734375 + x = -1.281953944 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.052734375' to each side of the equation. 0.052734375 + -0.052734375 + x = -1.281953944 + -0.052734375 Combine like terms: 0.052734375 + -0.052734375 = 0.000000000 0.000000000 + x = -1.281953944 + -0.052734375 x = -1.281953944 + -0.052734375 Combine like terms: -1.281953944 + -0.052734375 = -1.334688319 x = -1.334688319 Simplifying x = -1.334688319Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.229219569, -1.334688319}
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