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Simplifying 3x2 + 5 = (x + -2)(x + -5) Reorder the terms: 5 + 3x2 = (x + -2)(x + -5) Reorder the terms: 5 + 3x2 = (-2 + x)(x + -5) Reorder the terms: 5 + 3x2 = (-2 + x)(-5 + x) Multiply (-2 + x) * (-5 + x) 5 + 3x2 = (-2(-5 + x) + x(-5 + x)) 5 + 3x2 = ((-5 * -2 + x * -2) + x(-5 + x)) 5 + 3x2 = ((10 + -2x) + x(-5 + x)) 5 + 3x2 = (10 + -2x + (-5 * x + x * x)) 5 + 3x2 = (10 + -2x + (-5x + x2)) Combine like terms: -2x + -5x = -7x 5 + 3x2 = (10 + -7x + x2) Solving 5 + 3x2 = 10 + -7x + x2 Solving for variable 'x'. Reorder the terms: 5 + -10 + 7x + 3x2 + -1x2 = 10 + -7x + x2 + -10 + 7x + -1x2 Combine like terms: 5 + -10 = -5 -5 + 7x + 3x2 + -1x2 = 10 + -7x + x2 + -10 + 7x + -1x2 Combine like terms: 3x2 + -1x2 = 2x2 -5 + 7x + 2x2 = 10 + -7x + x2 + -10 + 7x + -1x2 Reorder the terms: -5 + 7x + 2x2 = 10 + -10 + -7x + 7x + x2 + -1x2 Combine like terms: 10 + -10 = 0 -5 + 7x + 2x2 = 0 + -7x + 7x + x2 + -1x2 -5 + 7x + 2x2 = -7x + 7x + x2 + -1x2 Combine like terms: -7x + 7x = 0 -5 + 7x + 2x2 = 0 + x2 + -1x2 -5 + 7x + 2x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 -5 + 7x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2.5 + 3.5x + x2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 3.5x + 2.5 + x2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 3.5x + x2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 3.5x + x2 = 0 + 2.5 3.5x + x2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 3.5x + x2 = 2.5 The x term is 3.5x. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5x + 3.0625 + x2 = 2.5 + 3.0625 Reorder the terms: 3.0625 + 3.5x + x2 = 2.5 + 3.0625 Combine like terms: 2.5 + 3.0625 = 5.5625 3.0625 + 3.5x + x2 = 5.5625 Factor a perfect square on the left side: (x + 1.75)(x + 1.75) = 5.5625 Calculate the square root of the right side: 2.358495283 Break this problem into two subproblems by setting (x + 1.75) equal to 2.358495283 and -2.358495283.Subproblem 1
x + 1.75 = 2.358495283 Simplifying x + 1.75 = 2.358495283 Reorder the terms: 1.75 + x = 2.358495283 Solving 1.75 + x = 2.358495283 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 2.358495283 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 2.358495283 + -1.75 x = 2.358495283 + -1.75 Combine like terms: 2.358495283 + -1.75 = 0.608495283 x = 0.608495283 Simplifying x = 0.608495283Subproblem 2
x + 1.75 = -2.358495283 Simplifying x + 1.75 = -2.358495283 Reorder the terms: 1.75 + x = -2.358495283 Solving 1.75 + x = -2.358495283 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -2.358495283 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -2.358495283 + -1.75 x = -2.358495283 + -1.75 Combine like terms: -2.358495283 + -1.75 = -4.108495283 x = -4.108495283 Simplifying x = -4.108495283Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.608495283, -4.108495283}
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