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Simplifying 4x + 7x2 = 76(4x + 6) Reorder the terms: 4x + 7x2 = 76(6 + 4x) 4x + 7x2 = (6 * 76 + 4x * 76) 4x + 7x2 = (456 + 304x) Solving 4x + 7x2 = 456 + 304x Solving for variable 'x'. Reorder the terms: -456 + 4x + -304x + 7x2 = 456 + 304x + -456 + -304x Combine like terms: 4x + -304x = -300x -456 + -300x + 7x2 = 456 + 304x + -456 + -304x Reorder the terms: -456 + -300x + 7x2 = 456 + -456 + 304x + -304x Combine like terms: 456 + -456 = 0 -456 + -300x + 7x2 = 0 + 304x + -304x -456 + -300x + 7x2 = 304x + -304x Combine like terms: 304x + -304x = 0 -456 + -300x + 7x2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -65.14285714 + -42.85714286x + x2 = 0 Move the constant term to the right: Add '65.14285714' to each side of the equation. -65.14285714 + -42.85714286x + 65.14285714 + x2 = 0 + 65.14285714 Reorder the terms: -65.14285714 + 65.14285714 + -42.85714286x + x2 = 0 + 65.14285714 Combine like terms: -65.14285714 + 65.14285714 = 0.00000000 0.00000000 + -42.85714286x + x2 = 0 + 65.14285714 -42.85714286x + x2 = 0 + 65.14285714 Combine like terms: 0 + 65.14285714 = 65.14285714 -42.85714286x + x2 = 65.14285714 The x term is -42.85714286x. Take half its coefficient (-21.42857143). Square it (459.1836735) and add it to both sides. Add '459.1836735' to each side of the equation. -42.85714286x + 459.1836735 + x2 = 65.14285714 + 459.1836735 Reorder the terms: 459.1836735 + -42.85714286x + x2 = 65.14285714 + 459.1836735 Combine like terms: 65.14285714 + 459.1836735 = 524.32653064 459.1836735 + -42.85714286x + x2 = 524.32653064 Factor a perfect square on the left side: (x + -21.42857143)(x + -21.42857143) = 524.32653064 Calculate the square root of the right side: 22.898177452 Break this problem into two subproblems by setting (x + -21.42857143) equal to 22.898177452 and -22.898177452.Subproblem 1
x + -21.42857143 = 22.898177452 Simplifying x + -21.42857143 = 22.898177452 Reorder the terms: -21.42857143 + x = 22.898177452 Solving -21.42857143 + x = 22.898177452 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '21.42857143' to each side of the equation. -21.42857143 + 21.42857143 + x = 22.898177452 + 21.42857143 Combine like terms: -21.42857143 + 21.42857143 = 0.00000000 0.00000000 + x = 22.898177452 + 21.42857143 x = 22.898177452 + 21.42857143 Combine like terms: 22.898177452 + 21.42857143 = 44.326748882 x = 44.326748882 Simplifying x = 44.326748882Subproblem 2
x + -21.42857143 = -22.898177452 Simplifying x + -21.42857143 = -22.898177452 Reorder the terms: -21.42857143 + x = -22.898177452 Solving -21.42857143 + x = -22.898177452 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '21.42857143' to each side of the equation. -21.42857143 + 21.42857143 + x = -22.898177452 + 21.42857143 Combine like terms: -21.42857143 + 21.42857143 = 0.00000000 0.00000000 + x = -22.898177452 + 21.42857143 x = -22.898177452 + 21.42857143 Combine like terms: -22.898177452 + 21.42857143 = -1.469606022 x = -1.469606022 Simplifying x = -1.469606022Solution
The solution to the problem is based on the solutions from the subproblems. x = {44.326748882, -1.469606022}
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