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Simplifying (6z + 7)(8z + 9) = (10z + 11)(12z + 13) Reorder the terms: (7 + 6z)(8z + 9) = (10z + 11)(12z + 13) Reorder the terms: (7 + 6z)(9 + 8z) = (10z + 11)(12z + 13) Multiply (7 + 6z) * (9 + 8z) (7(9 + 8z) + 6z * (9 + 8z)) = (10z + 11)(12z + 13) ((9 * 7 + 8z * 7) + 6z * (9 + 8z)) = (10z + 11)(12z + 13) ((63 + 56z) + 6z * (9 + 8z)) = (10z + 11)(12z + 13) (63 + 56z + (9 * 6z + 8z * 6z)) = (10z + 11)(12z + 13) (63 + 56z + (54z + 48z2)) = (10z + 11)(12z + 13) Combine like terms: 56z + 54z = 110z (63 + 110z + 48z2) = (10z + 11)(12z + 13) Reorder the terms: 63 + 110z + 48z2 = (11 + 10z)(12z + 13) Reorder the terms: 63 + 110z + 48z2 = (11 + 10z)(13 + 12z) Multiply (11 + 10z) * (13 + 12z) 63 + 110z + 48z2 = (11(13 + 12z) + 10z * (13 + 12z)) 63 + 110z + 48z2 = ((13 * 11 + 12z * 11) + 10z * (13 + 12z)) 63 + 110z + 48z2 = ((143 + 132z) + 10z * (13 + 12z)) 63 + 110z + 48z2 = (143 + 132z + (13 * 10z + 12z * 10z)) 63 + 110z + 48z2 = (143 + 132z + (130z + 120z2)) Combine like terms: 132z + 130z = 262z 63 + 110z + 48z2 = (143 + 262z + 120z2) Solving 63 + 110z + 48z2 = 143 + 262z + 120z2 Solving for variable 'z'. Reorder the terms: 63 + -143 + 110z + -262z + 48z2 + -120z2 = 143 + 262z + 120z2 + -143 + -262z + -120z2 Combine like terms: 63 + -143 = -80 -80 + 110z + -262z + 48z2 + -120z2 = 143 + 262z + 120z2 + -143 + -262z + -120z2 Combine like terms: 110z + -262z = -152z -80 + -152z + 48z2 + -120z2 = 143 + 262z + 120z2 + -143 + -262z + -120z2 Combine like terms: 48z2 + -120z2 = -72z2 -80 + -152z + -72z2 = 143 + 262z + 120z2 + -143 + -262z + -120z2 Reorder the terms: -80 + -152z + -72z2 = 143 + -143 + 262z + -262z + 120z2 + -120z2 Combine like terms: 143 + -143 = 0 -80 + -152z + -72z2 = 0 + 262z + -262z + 120z2 + -120z2 -80 + -152z + -72z2 = 262z + -262z + 120z2 + -120z2 Combine like terms: 262z + -262z = 0 -80 + -152z + -72z2 = 0 + 120z2 + -120z2 -80 + -152z + -72z2 = 120z2 + -120z2 Combine like terms: 120z2 + -120z2 = 0 -80 + -152z + -72z2 = 0 Factor out the Greatest Common Factor (GCF), '-8'. -8(10 + 19z + 9z2) = 0 Factor a trinomial. -8((10 + 9z)(1 + z)) = 0 Ignore the factor -8.Subproblem 1
Set the factor '(10 + 9z)' equal to zero and attempt to solve: Simplifying 10 + 9z = 0 Solving 10 + 9z = 0 Move all terms containing z to the left, all other terms to the right. Add '-10' to each side of the equation. 10 + -10 + 9z = 0 + -10 Combine like terms: 10 + -10 = 0 0 + 9z = 0 + -10 9z = 0 + -10 Combine like terms: 0 + -10 = -10 9z = -10 Divide each side by '9'. z = -1.111111111 Simplifying z = -1.111111111Subproblem 2
Set the factor '(1 + z)' equal to zero and attempt to solve: Simplifying 1 + z = 0 Solving 1 + z = 0 Move all terms containing z to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + z = 0 + -1 Combine like terms: 1 + -1 = 0 0 + z = 0 + -1 z = 0 + -1 Combine like terms: 0 + -1 = -1 z = -1 Simplifying z = -1Solution
z = {-1.111111111, -1}
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