| 1. |
info: 2025-04-26 01:38:18.539 RelationalEventId.CommandExecuted[20101] (Microsoft.EntityFrameworkCore.Database.Command)
19ms [Parameters=[@p7='8998d746-1759-46ff-b255-c397438cf3c4', @p0='True', @p1='False', @p2='1', @p3='b978d09b-77b0-444d-bb0f-385dc423e71a', @p4='59.jpg' (Size = 100), @p5='https://knowledgenest.blob.core.windows.net/scanners/0/[Matematika][Matematika 11 Konspektas ir užduočių bankas][Matematika 11 Konspektas ir užduočių bankas]/pages/59.jpg' (Size = 1000), @p6='""
## II. MODELIAI IR SĄRYŠIAI
**1 pavyzdys.** Išspręskime lygtį 2x³ + 54 = 0.
*Sprendimas.*
2x³ + 54 = 0,
x³ = -27,
x = -3.
*Atsakymas.* -3.
**2 pavyzdys.** Išspręskime lygtį 3x⁴ = 48.
*Sprendimas.*
3x⁴ = 48 (abi lygties puses dalijame iš 3),
x⁴ = 16 (iš abiejų lygties pusių traukiame ketvirtojo laipsnio šaknį),
x = ±2.
*Atsakymas.* -2; 2.
Laipsnines lygtis patogu spręsti ir skaidymo daugikliais būdu.
### Skaidymo daugikliais būdas
Taikoma sandaugos lygumo nuliui savybė: sandauga lygi nuliui tada ir tik tada, kai bent vienas daugiklis lygus nuliui, o kiti daugikliai su tomis nežinomojo reikšmėmis turi prasmę.
**Pavyzdys.** Išspręskime trečiojo laipsnio lygtį x³ − x² − 4(x − 1)² = 0.
*Sprendimas.*
x²(x - 1) - 4(x − 1)² = 0,
(x - 1)(x² - 4(x − 1)) = 0,
(x - 1)(x² - 4x + 4) = 0,
x - 1 = 0 arba x² - 4x + 4 = 0,
x₁ = 1; x₂,₃ = 2.
*Atsakymas.* 1; 2.
Skaidant lygtį daugikliais, daugiklių gali būti ir daugiau nei du. Tuomet kiekvieną jų prilyginame 0 ir gautąsias lygtis išsprendžiame.
### Bikvadratinės lygtys
ax⁴ + bx² + c = 0 (a, b, c – realieji skaičiai, a ≠ 0) pavidalo lygtis vadinama **bikvadratine lygtimi**.
Bikvadratinės lygties ax⁴ + bx² + c = 0 sprendimas.
1. Pažymime x² = t, tuomet x⁴ = t².
2. Gauname kvadratinę lygtį at² + bt + c = 0.
3. Apskaičiavę t reikšmę, grįžtame prie pažymėjimo x² = t ir apskaičiuojame x.
""' (Size = 4000)], CommandType='Text', CommandTimeout='30']
SET IMPLICIT_TRANSACTIONS OFF;
SET NOCOUNT ON;
UPDATE [ScannerPages] SET [Active] = @p0, [Deleted] = @p1, [Position] = @p2, [ScannerId] = @p3, [StorageMediaName] = @p4, [StorageMediaUrl] = @p5, [Text] = @p6
OUTPUT 1
WHERE [Id] = @p7;
|
| 2. |
info: 2025-04-26 01:38:19.524 RelationalEventId.CommandExecuted[20101] (Microsoft.EntityFrameworkCore.Database.Command)
8ms [Parameters=[@__id_0='8cca9677-7710-d3a1-5d50-8299a2e08223', @__site_Id_1='63d6eb4e-8e39-4ac6-8f69-55ee76feccce'], CommandType='Text', CommandTimeout='30']
SELECT [c].[Id], [c].[Active], [c].[Deleted], [c].[Description], [c].[DescriptionBackup], [c].[Left], [c].[ParentId], [c].[Position], [c].[ProcessorGenerateImagePrompt], [c].[Right], [c].[SiteId], [c].[StorageMediaName], [c].[StorageMediaUrl], [c].[Title], [c].[TitleBackup], [c].[TitleSlug], [c].[TotalChildren], [c].[TotalScanners], [c].[TotalYoutubeVideos]
FROM [Categories] AS [c] WITH (NOLOCK)
WHERE [c].[Id] = @__id_0 AND [c].[SiteId] = @__site_Id_1 AND [c].[Active] = CAST(1 AS bit) AND [c].[Deleted] = CAST(0 AS bit)
ORDER BY [c].[Id]
|
| 3. |
info: 2025-04-26 01:38:19.527 RelationalEventId.CommandExecuted[20101] (Microsoft.EntityFrameworkCore.Database.Command)
2ms [Parameters=[@__id_0='8cca9677-7710-d3a1-5d50-8299a2e08223', @__site_Id_1='63d6eb4e-8e39-4ac6-8f69-55ee76feccce'], CommandType='Text', CommandTimeout='30']
SELECT [c4].[Id], [c4].[Active], [c4].[Deleted], [c4].[Description], [c4].[DescriptionBackup], [c4].[Left], [c4].[ParentId], [c4].[Position], [c4].[ProcessorGenerateImagePrompt], [c4].[Right], [c4].[SiteId], [c4].[StorageMediaName], [c4].[StorageMediaUrl], [c4].[Title], [c4].[TitleBackup], [c4].[TitleSlug], [c4].[TotalChildren], [c4].[TotalScanners], [c4].[TotalYoutubeVideos], [c].[Id]
FROM [Categories] AS [c] WITH (NOLOCK)
INNER JOIN (
SELECT [c0].[Id], [c0].[Active], [c0].[Deleted], [c0].[Description], [c0].[DescriptionBackup], [c0].[Left], [c0].[ParentId], [c0].[Position], [c0].[ProcessorGenerateImagePrompt], [c0].[Right], [c0].[SiteId], [c0].[StorageMediaName], [c0].[StorageMediaUrl], [c0].[Title], [c0].[TitleBackup], [c0].[TitleSlug], [c0].[TotalChildren], [c0].[TotalScanners], [c0].[TotalYoutubeVideos]
FROM [Categories] AS [c0] WITH (NOLOCK)
WHERE [c0].[Active] = CAST(1 AS bit) AND [c0].[Deleted] = CAST(0 AS bit)
) AS [c4] ON [c].[Id] = [c4].[ParentId]
WHERE [c].[Id] = @__id_0 AND [c].[SiteId] = @__site_Id_1 AND [c].[Active] = CAST(1 AS bit) AND [c].[Deleted] = CAST(0 AS bit)
ORDER BY [c].[Id], [c4].[Position], [c4].[Id]
|
| 4. |
info: 2025-04-26 01:38:19.545 RelationalEventId.CommandExecuted[20101] (Microsoft.EntityFrameworkCore.Database.Command)
8ms [Parameters=[@__id_0='8cca9677-7710-d3a1-5d50-8299a2e08223', @__site_Id_1='63d6eb4e-8e39-4ac6-8f69-55ee76feccce'], CommandType='Text', CommandTimeout='30']
SELECT [c5].[Id], [c5].[Active], [c5].[Deleted], [c5].[Description], [c5].[DescriptionBackup], [c5].[Left], [c5].[ParentId], [c5].[Position], [c5].[ProcessorGenerateImagePrompt], [c5].[Right], [c5].[SiteId], [c5].[StorageMediaName], [c5].[StorageMediaUrl], [c5].[Title], [c5].[TitleBackup], [c5].[TitleSlug], [c5].[TotalChildren], [c5].[TotalScanners], [c5].[TotalYoutubeVideos], [c].[Id], [c4].[Id]
FROM [Categories] AS [c] WITH (NOLOCK)
INNER JOIN (
SELECT [c0].[Id], [c0].[ParentId], [c0].[Position]
FROM [Categories] AS [c0] WITH (NOLOCK)
WHERE [c0].[Active] = CAST(1 AS bit) AND [c0].[Deleted] = CAST(0 AS bit)
) AS [c4] ON [c].[Id] = [c4].[ParentId]
INNER JOIN (
SELECT [c1].[Id], [c1].[Active], [c1].[Deleted], [c1].[Description], [c1].[DescriptionBackup], [c1].[Left], [c1].[ParentId], [c1].[Position], [c1].[ProcessorGenerateImagePrompt], [c1].[Right], [c1].[SiteId], [c1].[StorageMediaName], [c1].[StorageMediaUrl], [c1].[Title], [c1].[TitleBackup], [c1].[TitleSlug], [c1].[TotalChildren], [c1].[TotalScanners], [c1].[TotalYoutubeVideos]
FROM [Categories] AS [c1] WITH (NOLOCK)
WHERE [c1].[Active] = CAST(1 AS bit) AND [c1].[Deleted] = CAST(0 AS bit)
) AS [c5] ON [c4].[Id] = [c5].[ParentId]
WHERE [c].[Id] = @__id_0 AND [c].[SiteId] = @__site_Id_1 AND [c].[Active] = CAST(1 AS bit) AND [c].[Deleted] = CAST(0 AS bit)
ORDER BY [c].[Id], [c4].[Position], [c4].[Id], [c5].[Position]
|
| 5. |
info: 2025-04-26 01:38:19.583 RelationalEventId.CommandExecuted[20101] (Microsoft.EntityFrameworkCore.Database.Command)
29ms [Parameters=[@__id_0='8cca9677-7710-d3a1-5d50-8299a2e08223', @__site_Id_1='63d6eb4e-8e39-4ac6-8f69-55ee76feccce'], CommandType='Text', CommandTimeout='30']
SELECT [s3].[ScannerId], [s3].[CategoryId], [s3].[Id], [s3].[Active], [s3].[CheatSheet], [s3].[CheatSheet2], [s3].[Deleted], [s3].[FactErrors], [s3].[FactErrorsExplanation], [s3].[ParentCategory2Id], [s3].[ParentCategoryId], [s3].[ProcessorGenerateImagePrompt], [s3].[Prompt], [s3].[Reference], [s3].[ReferenceSlug], [s3].[SiteId], [s3].[SpellingErrors], [s3].[SpellingErrorsExplanation], [s3].[Summary], [s3].[Title], [s3].[TitleBackup], [s3].[TitleSlug], [s3].[TotalCards], [s3].[TotalCards2], [s3].[TotalCategories], [s3].[TotalPages], [s3].[Verified], [c].[Id], [c4].[Id]
FROM [Categories] AS [c] WITH (NOLOCK)
INNER JOIN (
SELECT [c0].[Id], [c0].[ParentId], [c0].[Position]
FROM [Categories] AS [c0] WITH (NOLOCK)
WHERE [c0].[Active] = CAST(1 AS bit) AND [c0].[Deleted] = CAST(0 AS bit)
) AS [c4] ON [c].[Id] = [c4].[ParentId]
INNER JOIN (
SELECT [c2].[ScannerId], [c2].[CategoryId], [s].[Id], [s].[Active], [s].[CheatSheet], [s].[CheatSheet2], [s].[Deleted], [s].[FactErrors], [s].[FactErrorsExplanation], [s].[ParentCategory2Id], [s].[ParentCategoryId], [s].[ProcessorGenerateImagePrompt], [s].[Prompt], [s].[Reference], [s].[ReferenceSlug], [s].[SiteId], [s].[SpellingErrors], [s].[SpellingErrorsExplanation], [s].[Summary], [s].[Title], [s].[TitleBackup], [s].[TitleSlug], [s].[TotalCards], [s].[TotalCards2], [s].[TotalCategories], [s].[TotalPages], [s].[Verified]
FROM [CategoryScanners] AS [c2]
INNER JOIN [Scanners] AS [s] ON [c2].[ScannerId] = [s].[Id]
WHERE [s].[Active] = CAST(1 AS bit) AND [s].[Deleted] = CAST(0 AS bit)
) AS [s3] ON [c4].[Id] = [s3].[CategoryId]
WHERE [c].[Id] = @__id_0 AND [c].[SiteId] = @__site_Id_1 AND [c].[Active] = CAST(1 AS bit) AND [c].[Deleted] = CAST(0 AS bit)
ORDER BY [c].[Id], [c4].[Position], [c4].[Id], [s3].[Id], [s3].[ScannerId], [s3].[CategoryId]
|
| 6. |
info: 2025-04-26 01:38:19.708 RelationalEventId.CommandExecuted[20101] (Microsoft.EntityFrameworkCore.Database.Command)
29ms [Parameters=[@__id_0='8cca9677-7710-d3a1-5d50-8299a2e08223', @__site_Id_1='63d6eb4e-8e39-4ac6-8f69-55ee76feccce'], CommandType='Text', CommandTimeout='30']
SELECT [s4].[Id], [s4].[Active], [s4].[Context], [s4].[ContextBackup], [s4].[ContextStorageMediaName], [s4].[ContextStorageMediaUrl], [s4].[Deleted], [s4].[FactErrors], [s4].[FactErrorsExplanation], [s4].[Facts], [s4].[FactsBackup], [s4].[FactsStorageMediaName], [s4].[FactsStorageMediaUrl], [s4].[Position], [s4].[ProcessorGenerateImagePrompt], [s4].[Prompt], [s4].[ScannerId], [s4].[SpellingErrors], [s4].[SpellingErrorsExplanation], [s4].[StorageMediaName], [s4].[StorageMediaUrl], [s4].[Summary], [s4].[SummaryBackup], [s4].[SummaryStorageMediaName], [s4].[SummaryStorageMediaUrl], [s4].[Title], [s4].[TitleSlug], [s4].[TotalFillTheBlankSelectionQuestions], [s4].[TotalMultipleSelectionQuestions], [s4].[TotalOrderingQuestions], [s4].[TotalQuestionPoints], [s4].[TotalQuestions], [s4].[TotalShortAnswerQuestions], [s4].[TotalSingleSelectionQuestions], [s4].[TotalSources], [s4].[TotalTrueFalseQuestions], [s4].[Verified], [s4].[Version], [c].[Id], [c4].[Id], [s3].[ScannerId], [s3].[CategoryId], [s3].[Id]
FROM [Categories] AS [c] WITH (NOLOCK)
INNER JOIN (
SELECT [c0].[Id], [c0].[ParentId], [c0].[Position]
FROM [Categories] AS [c0] WITH (NOLOCK)
WHERE [c0].[Active] = CAST(1 AS bit) AND [c0].[Deleted] = CAST(0 AS bit)
) AS [c4] ON [c].[Id] = [c4].[ParentId]
INNER JOIN (
SELECT [c2].[ScannerId], [c2].[CategoryId], [s].[Id]
FROM [CategoryScanners] AS [c2]
INNER JOIN [Scanners] AS [s] ON [c2].[ScannerId] = [s].[Id]
WHERE [s].[Active] = CAST(1 AS bit) AND [s].[Deleted] = CAST(0 AS bit)
) AS [s3] ON [c4].[Id] = [s3].[CategoryId]
INNER JOIN (
SELECT [s0].[Id], [s0].[Active], [s0].[Context], [s0].[ContextBackup], [s0].[ContextStorageMediaName], [s0].[ContextStorageMediaUrl], [s0].[Deleted], [s0].[FactErrors], [s0].[FactErrorsExplanation], [s0].[Facts], [s0].[FactsBackup], [s0].[FactsStorageMediaName], [s0].[FactsStorageMediaUrl], [s0].[Position], [s0].[ProcessorGenerateImagePrompt], [s0].[Prompt], [s0].[ScannerId], [s0].[SpellingErrors], [s0].[SpellingErrorsExplanation], [s0].[StorageMediaName], [s0].[StorageMediaUrl], [s0].[Summary], [s0].[SummaryBackup], [s0].[SummaryStorageMediaName], [s0].[SummaryStorageMediaUrl], [s0].[Title], [s0].[TitleSlug], [s0].[TotalFillTheBlankSelectionQuestions], [s0].[TotalMultipleSelectionQuestions], [s0].[TotalOrderingQuestions], [s0].[TotalQuestionPoints], [s0].[TotalQuestions], [s0].[TotalShortAnswerQuestions], [s0].[TotalSingleSelectionQuestions], [s0].[TotalSources], [s0].[TotalTrueFalseQuestions], [s0].[Verified], [s0].[Version]
FROM [ScannerCards] AS [s0]
WHERE [s0].[Active] = CAST(1 AS bit) AND [s0].[Deleted] = CAST(0 AS bit)
) AS [s4] ON [s3].[Id] = [s4].[ScannerId]
WHERE [c].[Id] = @__id_0 AND [c].[SiteId] = @__site_Id_1 AND [c].[Active] = CAST(1 AS bit) AND [c].[Deleted] = CAST(0 AS bit)
ORDER BY [c].[Id], [c4].[Position], [c4].[Id], [s3].[Id], [s3].[ScannerId], [s3].[CategoryId], [s4].[TitleSlug]
|
| 7. |
info: 2025-04-26 01:38:19.729 RelationalEventId.CommandExecuted[20101] (Microsoft.EntityFrameworkCore.Database.Command)
8ms [Parameters=[@__id_0='8cca9677-7710-d3a1-5d50-8299a2e08223', @__site_Id_1='63d6eb4e-8e39-4ac6-8f69-55ee76feccce'], CommandType='Text', CommandTimeout='30']
SELECT [s5].[ScannerId], [s5].[CategoryId], [s5].[Id], [s5].[Active], [s5].[CheatSheet], [s5].[CheatSheet2], [s5].[Deleted], [s5].[FactErrors], [s5].[FactErrorsExplanation], [s5].[ParentCategory2Id], [s5].[ParentCategoryId], [s5].[ProcessorGenerateImagePrompt], [s5].[Prompt], [s5].[Reference], [s5].[ReferenceSlug], [s5].[SiteId], [s5].[SpellingErrors], [s5].[SpellingErrorsExplanation], [s5].[Summary], [s5].[Title], [s5].[TitleBackup], [s5].[TitleSlug], [s5].[TotalCards], [s5].[TotalCards2], [s5].[TotalCategories], [s5].[TotalPages], [s5].[Verified], [c].[Id]
FROM [Categories] AS [c] WITH (NOLOCK)
INNER JOIN (
SELECT [c3].[ScannerId], [c3].[CategoryId], [s1].[Id], [s1].[Active], [s1].[CheatSheet], [s1].[CheatSheet2], [s1].[Deleted], [s1].[FactErrors], [s1].[FactErrorsExplanation], [s1].[ParentCategory2Id], [s1].[ParentCategoryId], [s1].[ProcessorGenerateImagePrompt], [s1].[Prompt], [s1].[Reference], [s1].[ReferenceSlug], [s1].[SiteId], [s1].[SpellingErrors], [s1].[SpellingErrorsExplanation], [s1].[Summary], [s1].[Title], [s1].[TitleBackup], [s1].[TitleSlug], [s1].[TotalCards], [s1].[TotalCards2], [s1].[TotalCategories], [s1].[TotalPages], [s1].[Verified]
FROM [CategoryScanners] AS [c3]
INNER JOIN [Scanners] AS [s1] ON [c3].[ScannerId] = [s1].[Id]
WHERE [s1].[Active] = CAST(1 AS bit) AND [s1].[Deleted] = CAST(0 AS bit)
) AS [s5] ON [c].[Id] = [s5].[CategoryId]
WHERE [c].[Id] = @__id_0 AND [c].[SiteId] = @__site_Id_1 AND [c].[Active] = CAST(1 AS bit) AND [c].[Deleted] = CAST(0 AS bit)
ORDER BY [c].[Id], [s5].[Id], [s5].[ScannerId], [s5].[CategoryId]
|
| 8. |
info: 2025-04-26 01:38:19.959 RelationalEventId.CommandExecuted[20101] (Microsoft.EntityFrameworkCore.Database.Command)
15ms [Parameters=[@__site_Id_0='63d6eb4e-8e39-4ac6-8f69-55ee76feccce', @__item_Left_1='709', @__item_Right_2='726'], CommandType='Text', CommandTimeout='30']
SELECT [c].[Id], [c].[Active], [c].[Deleted], [c].[Description], [c].[DescriptionBackup], [c].[Left], [c].[ParentId], [c].[Position], [c].[ProcessorGenerateImagePrompt], [c].[Right], [c].[SiteId], [c].[StorageMediaName], [c].[StorageMediaUrl], [c].[Title], [c].[TitleBackup], [c].[TitleSlug], [c].[TotalChildren], [c].[TotalScanners], [c].[TotalYoutubeVideos]
FROM [Categories] AS [c] WITH (NOLOCK)
WHERE [c].[SiteId] = @__site_Id_0 AND [c].[Left] <= @__item_Left_1 AND [c].[Right] >= @__item_Right_2 AND [c].[Deleted] = CAST(0 AS bit)
ORDER BY [c].[Left]
|
| 118 ms |
