Borrowings - Schedule of Long-term Debt (Details) - USD ($)	Sep. 30, 2022	Sep. 30, 2021
Debt Instrument [Line Items]
Unamortized debt issuance costs	$ (7,280,000)	$ (8,983,000)
Total debt, net	983,440,000	1,054,030,000
Revolving Credit Facility
Debt Instrument [Line Items]
Secured Revolving Credit Facility	0	0
Senior Notes
Debt Instrument [Line Items]
Senior Unsecured Term Loan	0	50,000,000
Unamortized debt issuance costs	(7,280,000)	(8,983,000)
Total debt, net	911,170,000	983,827,000
Junior Subordinated Notes
Debt Instrument [Line Items]
Total debt, net	72,270,000	70,203,000
Unamortized discount	28,503,000	30,570,000
6.750% Senior Notes (2025 Notes) \| Senior Notes
Debt Instrument [Line Items]
Total debt, net	$ 211,195,000	229,555,000
Stated interest rate on debt instrument	6.75%
5.875% Senior Notes (2027 Notes) \| Senior Notes
Debt Instrument [Line Items]
Total debt, net	$ 357,255,000	363,255,000
Stated interest rate on debt instrument	5.875%
7.250% Senior Notes (2029 Notes) \| Senior Notes
Debt Instrument [Line Items]
Total debt, net	$ 350,000,000	$ 350,000,000
Stated interest rate on debt instrument	7.25%
Junior Subordinated Notes
Debt Instrument [Line Items]
Unamortized discount	$ 28,500,000

Contractual interest rate for funds borrowed, under the debt agreement.

Reference 1: http://www.xbrl.org/2003/role/disclosureRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 470
-SubTopic 20
-Section 50
-Paragraph 1B
-Subparagraph (b)
-URI https://asc.fasb.org/extlink&oid=123466505&loc=SL123495323-112611

Reference 2: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 210
-SubTopic 10
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02.22(a)(1))
-URI https://asc.fasb.org/extlink&oid=120391452&loc=d3e13212-122682

+ Details

Name:	us-gaap_DebtInstrumentInterestRateStatedPercentage
Namespace Prefix:	us-gaap_
Data Type:	dtr-types:percentItemType
Balance Type:	na
Period Type:	instant

- Definition

Line items represent financial concepts included in a table. These concepts are used to disclose reportable information associated with domain members defined in one or many axes to the table.

+ References

No definition available.

+ Details

Name:	us-gaap_DebtInstrumentLineItems
Namespace Prefix:	us-gaap_
Data Type:	xbrli:stringItemType
Balance Type:	na
Period Type:	duration

- Definition

Amount, after accumulated amortization, of debt discount.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 835
-SubTopic 30
-Section 45
-Paragraph 1A
-URI https://asc.fasb.org/extlink&oid=124435984&loc=d3e28541-108399

Reference 2: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 835
-SubTopic 30
-Section 55
-Paragraph 8
-URI https://asc.fasb.org/extlink&oid=114775985&loc=d3e28878-108400

Reference 3: http://www.xbrl.org/2009/role/commonPracticeRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 470
-SubTopic 20
-Section 50
-Paragraph 1D
-Subparagraph (a)
-URI https://asc.fasb.org/extlink&oid=123466505&loc=SL123495340-112611

+ Details

Name:	us-gaap_DebtInstrumentUnamortizedDiscount
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

- Definition

Represents the aggregate of total long-term debt, including current maturities and short-term debt.

+ References

No definition available.

+ Details

Name:	us-gaap_DebtLongtermAndShorttermCombinedAmount
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Definition

Amount of current borrowing capacity under the credit facility considering any current restrictions on the amount that could be borrowed (for example, borrowings may be limited by the amount of current assets), but without considering any amounts currently outstanding under the facility.

+ References

+ Details

Name:	us-gaap_LineOfCreditFacilityCurrentBorrowingCapacity
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Definition

The remaining balance of debt issuance expenses that were capitalized and are being amortized against income over the lives of the respective bond issues. This does not include the amounts capitalized as part of the cost of the utility plant or asset.

+ References

No definition available.

+ Details

Name:	us-gaap_UnamortizedDebtIssuanceExpense
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	instant

- Definition

Including the current and noncurrent portions, carrying value as of the balance sheet date of uncollateralized debt obligations (with maturities initially due after one year or beyond the operating cycle if longer).

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 210
-SubTopic 10
-Section S99
-Paragraph 1
-Subparagraph (SX 210.5-02(22))
-URI https://asc.fasb.org/extlink&oid=120391452&loc=d3e13212-122682

Reference 2: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 942
-SubTopic 210
-Section S99
-Paragraph 1
-Subparagraph (SX 210.9-03(16))
-URI https://asc.fasb.org/extlink&oid=126897435&loc=d3e534808-122878

Reference 3: http://fasb.org/us-gaap/role/ref/legacyRef
-Publisher FASB
-Name Accounting Standards Codification
-Topic 944
-SubTopic 210
-Section S99
-Paragraph 1
-Subparagraph (SX 210.7-03(a)(16))
-URI https://asc.fasb.org/extlink&oid=126734703&loc=d3e572229-122910

+ Details

Name:	us-gaap_UnsecuredDebt
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	credit
Period Type:	instant

- Details

Name:	us-gaap_CreditFacilityAxis=us-gaap_RevolvingCreditFacilityMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	bzh_LongtermandShorttermDebtTypeAxis=us-gaap_SeniorNotesMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	bzh_LongtermandShorttermDebtTypeAxis=us-gaap_JuniorSubordinatedDebtMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_DebtInstrumentAxis=bzh_A634SeniorNotesMaturingMarchof2025Member
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_DebtInstrumentAxis=bzh_A578SeniorNotesMaturingOctober2027Member
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_DebtInstrumentAxis=bzh_A714SeniorNotesMaturingOctober2029Member
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_LongtermDebtTypeAxis=us-gaap_JuniorSubordinatedDebtMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type: