Mokhoa oa ho rarolla li-quadratic equation

# Mokhoa oa ho Rarolla Li-Quadratic Equations

Li-equation tsa quadratic ke e 'ngoe ea mefuta ea motheo le e atisang ho fumanoa ea li-equation tsa algebraic lipalo. Equation ena e na le sebopeho se akaretsang \( ax^2 + bx + c = 0 \), moo \( a \), \( b \), le \( c \) e leng li-constant, 'me \( x \) ke phetoho eo boleng ba eona bo lokelang ho fumanoa. Sehloohong sena, re tla tšohla litsela tse fapaneng tsa ho rarolla li-equation tsa quadratic, ho kenyeletsoa mekhoa ea ho etsa li-factoring, ho sebelisa foromo ea quadratic, ho tlatsa sekwere, le mekhoa ea litšoantšo.

## 1. Mokhoa oa ho Factoring

E 'ngoe ea litsela tse bonolo ka ho fetisisa tsa ho rarolla equation ea quadratic ke ho e bala ka mokhoa o nepahetseng. Leha ho le joalo, mokhoa ona o sebetsa feela haeba equation ea quadratic e ka baloa ka mokhoa o bonolo.

### Mehato:

1. Etsa bonnete ba hore equation e ka sebopeho se tloaelehileng:
Tekanyo ea quadratic e tlameha ho ba ka sebopeho sa \( ax^2 + bx + c = 0 \).

2. Fumana linomoro tse peli tseo ha li atisitsoe li fanang ka \( ac \) (sehlahisoa sa \( a \) le \( c \)) le ha li eketsoa li fanang ka \( b \):
Mohlala, haeba equation e le \( x^2 + 5x + 6 = 0 \), re batla dinomoro tse pedi tseo ha di atiswa di etsang 6 mme ha di eketswa di etsang 5. Dinomoro tseo ke 2 le 3.

3. Arola dipalo tse pedi ka di-binomial tse pedi:
Tekanyo e ka hodimo e ka arolwa ka \( (x + 2)(x + 3) = 0 \).

4. Sebelisa molao-motheo oa sehlahisoa sa lefela:
Haeba \( (x + 2)(x + 3) = 0 \), ntlha e le 'ngoe kapa ka bobeli e lokela ho ba lefela. Kahoo, \( x + 2 = 0 \) kapa \( x + 3 = 0 \), e hlahisang \( x = -2 \) le \( x = -3 \).

Contoh:
– A re re re na le equation \( x^2 + 6x + 9 = 0 \).
– Re batla dinomoro tse pedi tseo ha di atiswa di fellang ka 9 mme ha di eketswa di fellang ka 6. Dinomoro tsena ke 3 le 3.
– Kahoo, equation e ka aroloa ka \( (x + 3)^2 = 0 \),
– Kahoo, re fumana \( x = -3 \).

## 2. Ho Sebelisa Foromo ea Quadratic

Haeba equation ea quadratic e ke ke ea lekanngoa habonolo, re ka sebelisa foromo ea quadratic. Foromo ea quadratic ke mokhoa o akaretsang o sebetsang ho li-equation tsohle tsa quadratic.

### Foromo:

\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]

### Mehato:

1. Hlalosa boleng ba \( a \), \( b \), le \( c \):
Ho tsoa ho equation \( ax^2 + bx + c = 0 \), hlwaya boleng ba \( a \), \( b \), le \( c \).

2. Kenya boleng bona sebakeng sa foromo ea quadratic:
Sebelisa foromo \( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \) ho fumana boleng ba \( x \).

3. Bala boleng ba khethollo (\( \Delta \)):
Khethollo ke \( b^2 – 4ac \).
– Haeba \( \Delta > 0 \), ho na le ditharollo tse pedi tse fapaneng.
– Haeba \( \Delta = 0 \), ho na le tharollo e le 'ngoe (motso o mobedi).
– Haeba \( \Delta < 0 \), ha ho na tharollo ea 'nete.

Mohlala: - A re re re na le equation \( 2x^2 + 4x - 6 = 0 \). - Kahoo, \( a = 2 \), \( b = 4 \), le \( c = -6 \). - Kenya boleng bona sebakeng sa foromo: \( x = \frac{-4 \pm \sqrt{16 + 48}}{4} \). - U tla fumana litharollo tse peli bakeng sa \( x \). ## 3. Ho Qetella Mokhoa oa Sekwere Ho Qetella mokhoa oa sekwere le hona ke mokhoa o tloaelehileng o sebelisoang ho rarolla li-quadratic equation, haholo-holo ha re batla ho utloisisa mohopolo oa li-square tse phethahetseng ka botebo. ### Mehato: 1. Netefatsa hore \( a = 1 \): Haeba \( a \neq 1 \), arola li-coefficient tsohle ka \( a \). 2. Suthisetsa constant ka lehlakoreng le letona la equation: A re re equation ea pele ke \( ax^2 + bx + c = 0 \). Kamora ho arola ka \( a \), e fetoha \( x^2 + \frac{b}{a}x = -\frac{c}{a} \). 3. Eketsa le ho tlosa \((\frac{b}{2a})^2 \) ka lehlakoreng le letšehali: Sena se etsa hore lehlakore le letšehali e be sekwere se phethahetseng. 4. Ngola equation e le sekwere se phethahetseng mme o rarolle: Etsa equation e le \((x + \frac{b}{2a})^2 = d \). Ebe, \( x + \frac{b}{2a} = \pm\sqrt{d} \), mme qetellong o rarolle bakeng sa \( x \). Mohlala: - Equation eo re batlang ho e rarolla ke \( x^2 + 6x + 5 = 0 \). - Re fetisetsa constant lehlakoreng le letona: \( x^2 + 6x = -5 \). - Eketsa le ho tlosa \( 9 \) (boleng ba \((\frac{6}{2})^2 \)) ka lehlakoreng le letshehadi: \( x^2 + 6x + 9 = 4 \), - Kahoo, equation jwale e fetoha \( (x + 3)^2 = 4 \). - Kahoo \( x + 3 = \pm 2 \), - Ka hona, \( x = -1 \) kapa \( x = -5 \). ## 4. Mokhoa oa Litšoantšo Mokhoa oa litšoantšo o kenyelletsa ho taka setšoantšo sa mosebetsi oa quadratic le ho bona moo o kopanang le x-axis. ### Mehato: 1. Theha mosebetsi oa quadratic \( y = ax^2 + bx + c \): Fetola equation ea quadratic ho ea mosebetsing \( y \) ka ho nkela 0 sebaka ka \( y \). 2. Kenya setšoantšo sa mosebetsi: Sebelisa boleng bo bong bakeng sa \( x \) ho taka setšoantšo sa parabola. 3. Batla li-x-intercepts: Lintlha tseo graf e kopanang le x-axis ho tsona ke litharollo tsa equation ea quadratic. Mohlala: - Nka \( x^2 - 3x + 2 = 0 \). - E fetole ho \( y = x^2 - 3x + 2 \). - Kenya setšoantšo sa mosebetsi. U tla bona hore graf e kopana le x-axis lintlheng \( x = 1 \) le \( x = 2 \). ## Qetello Ho rarolla di-quadratic equation ho ka etswa ka ho sebedisa mekgwa e fapaneng, jwalo ka ho etsa di-factoring, foromo ya quadratic, ho tlatsa sekwere, le mekgwa ya ditshwantsho. Ka ho utlwisisa le ho leka mokgwa o mong le o mong, re ka kgetha mokgwa o tshwanelang boemo kapa mofuta wa equation eo re tobaneng le yona. Re tshepa hore sengoloa sena se tla o thusa ho utlwisisa le ho rarolla di-quadratic equation hamolemo.