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Simplifying 42 = (x + 4)(2x + -7) Reorder the terms: 42 = (4 + x)(2x + -7) Reorder the terms: 42 = (4 + x)(-7 + 2x) Multiply (4 + x) * (-7 + 2x) 42 = (4(-7 + 2x) + x(-7 + 2x)) 42 = ((-7 * 4 + 2x * 4) + x(-7 + 2x)) 42 = ((-28 + 8x) + x(-7 + 2x)) 42 = (-28 + 8x + (-7 * x + 2x * x)) 42 = (-28 + 8x + (-7x + 2x2)) Combine like terms: 8x + -7x = 1x 42 = (-28 + 1x + 2x2) Solving 42 = -28 + 1x + 2x2 Solving for variable 'x'. Combine like terms: 42 + 28 = 70 70 + -1x + -2x2 = -28 + 1x + 2x2 + 28 + -1x + -2x2 Reorder the terms: 70 + -1x + -2x2 = -28 + 28 + 1x + -1x + 2x2 + -2x2 Combine like terms: -28 + 28 = 0 70 + -1x + -2x2 = 0 + 1x + -1x + 2x2 + -2x2 70 + -1x + -2x2 = 1x + -1x + 2x2 + -2x2 Combine like terms: 1x + -1x = 0 70 + -1x + -2x2 = 0 + 2x2 + -2x2 70 + -1x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 70 + -1x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -35 + 0.5x + x2 = 0 Move the constant term to the right: Add '35' to each side of the equation. -35 + 0.5x + 35 + x2 = 0 + 35 Reorder the terms: -35 + 35 + 0.5x + x2 = 0 + 35 Combine like terms: -35 + 35 = 0 0 + 0.5x + x2 = 0 + 35 0.5x + x2 = 0 + 35 Combine like terms: 0 + 35 = 35 0.5x + x2 = 35 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 35 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 35 + 0.0625 Combine like terms: 35 + 0.0625 = 35.0625 0.0625 + 0.5x + x2 = 35.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 35.0625 Calculate the square root of the right side: 5.921359641 Break this problem into two subproblems by setting (x + 0.25) equal to 5.921359641 and -5.921359641.Subproblem 1
x + 0.25 = 5.921359641 Simplifying x + 0.25 = 5.921359641 Reorder the terms: 0.25 + x = 5.921359641 Solving 0.25 + x = 5.921359641 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 5.921359641 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 5.921359641 + -0.25 x = 5.921359641 + -0.25 Combine like terms: 5.921359641 + -0.25 = 5.671359641 x = 5.671359641 Simplifying x = 5.671359641Subproblem 2
x + 0.25 = -5.921359641 Simplifying x + 0.25 = -5.921359641 Reorder the terms: 0.25 + x = -5.921359641 Solving 0.25 + x = -5.921359641 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -5.921359641 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -5.921359641 + -0.25 x = -5.921359641 + -0.25 Combine like terms: -5.921359641 + -0.25 = -6.171359641 x = -6.171359641 Simplifying x = -6.171359641Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.671359641, -6.171359641}
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