Kugadzira Mabasa eQuadratic: Gwaro Rakakwana
Pendauluan
Mumasvomhu, mabasa equadratic inyaya huru inowanzova hwaro hwekudzidza zvakawanda, kusanganisira calculus ne linear algebra. Kushandiswa kwemabasa equadratic kunopfuura dzidziso uye kunopinda mumhando dzakasiyana dzemashandisirwo anoshanda, kubva kufizikisi neinjiniya yemakanika kusvika kuhupfumi. Chinyorwa chino chichakurukura maitiro ekuvaka mabasa equadratic zvakadzama, kusanganisira tsananguro yawo, chimiro chakajairika, mhinduro dzemidzi, magirafu, uye mashandisirwo.
Kunzwisisa Mabasa eQuadratic
Basa requadratic ibasa rechipiri repolynomial, rinogona kuratidzwa muchimiro chakajairika:
\[ f(x) = ax^2 + bx + c \]
apo \(a\), \(b\), uye \(c\) zviri coefficients dzinogara dziripo, uye \(a \neq 0\) inovimbisa kuti basa racho ibasa re quadratic zvechokwadi. Fomu iri ndiro fomu rakajairika rebasa re quadratic.
Mamwe Mafomu eMabasa eQuadratic
Tisati taenderera mberi, zvakakosha kunzwisisa kuti kune nzira dzakawanda dzekuratidza basa re quadratic kunze kwechimiro chakajairika. Heano mamwe mafomu maviri anowanzo shandiswa:
1. Fomu reKuisa Mafactorization
Mabasa equadratic anogonawo kuratidzwa muchimiro che factorized, kunyanya kana midzi ichizivikanwa:
\[ f(x) = a(x – x_1)(x – x_2) \]
apo \(x_1\) uye \(x_2\) ndiwo midzi yebasa. Iyi nzira yekufactorization inobatsira zvikuru kana tatoziva mhinduro yebasa racho.
2. Chimiro cheVertex (Nhamba yepamusoro)
Basa re quadratic rinogonawo kushandurwa kuita vertex form, inova:
\[ f(x) = a(x – h)^2 + k \]
apo \((h, k)\) ari macoordinates e vertex ye parabola. Chimiro ichi chinobatsira zvikuru kana tichida kuziva nzvimbo uye chimiro che parabola.
Kugadzirisa Mabasa eQuadratic
Kuti tigadzirise kana kuwana mhinduro (midzi) dzebasa re quadratic \(ax^2 + bx + c = 0\), tinogona kushandisa nzira dzakasiyana-siyana, kusanganisira factorization, kupedzisa square, uye quadratic formula.
1. Kuenzanisa zvinhu
Nzira ye "factorization" inosanganisira kunyora patsva basa re "quadratic" maererano ne "product" ye "binomial numbers" mbiri:
\[ ax^2 + bx + c = a(x – x_1)(x – x_2) \]
Semuenzaniso, basa \(x^2 – 5x + 6 = 0\) rinogona kuverengerwa mu \((x – 2)(x – 3) = 0\), saka midzi yacho \(x = 2\) uye \(x = 3\).
2. Kupedzisa Chikwere
Nzira iyi inosanganisira kuwedzera nekubvisa kukosha kuti uchinje fomu rakajairika kuita fomu rakakwana rechikwere:
1. Tanga kubva pachimiro chakajairika: \(ax^2 + bx + c\).
2. Govanisa zvese ne \(a\) (kana \(a \neq 1\)).
3. Fambisa chinoramba chichienderana \(c/a\) kurudyi rweequation.
4. Wedzera uye bvisa \((b/2a)^2\).
5. Ronga divi rekuruboshwe woita kuti divi rerudyi rive nyore.
Semuenzaniso, yebasa \(x^2 + 6x + 8 = 0\):
\[x^2 + 6x = -8 \\
x^2 + 6x + 9 = 1 \\
(x + 3)^2 = 1 \\
x + 3 = \pm 1 \]
iyo inopa mhinduro \(x = -2\) uye \(x = -4\).
3. Fomura yeQuadratic
Nzira ye "quadratic" ndiyo nzira yakajairika uye yakavimbika yekuwana midzi yebasa re "quadratic":
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
Tichishandisa fomura iyi, tinogona kuwana midzi yechero basa re quadratic, kunyangwe kana kugadzirisa kana kupedzisa sikweya kusingashande. Semuenzaniso, kugadzirisa \(2x^2 + 4x – 6 = 0\):
\[ x = \frac{-4 \pm \sqrt{4^2 – 4 \cdot 2 \cdot (-6)}}{2 \cdot 2} \\
x = \frac{-4 \pm \sqrt{16 + 48}}{4} \\
x = \frac{-4 \pm \sqrt{64}}{4} \\
x = \frac{-4 \pm 8}{4} \]
Saka tinowana mhinduro mbiri: \(x = 1\) uye \(x = -3\).
Girafu yeMashandiro eQuadratic
Girafu yebasa re quadratic i parabola. Parabola iyi inogona kuvhurika kumusoro kana pasi zvichienderana nekukosha kwe coefficient \(a\):
– Kana \(a > 0\), parabola inovhurika kumusoro.
– Kana \(a < 0\), parabola inovhurika ichidzika. 1. Vertex neAxis yeSymmetry Vertex yeparabola (\(h, k\)) ndiyo poindi yepamusoro kana kuti yepasi pebasa requadratic. Macoordinates evertex \(h\) anogona kuwanikwa nefomura: \[ h = \frac{-b}{2a} \] Kuti tiwane \(k\), tinotsiva kukosha kwe \(h\) mu quadratic function \( f(h) = k \). Semuenzaniso, ye \(f(x) = 2x^2 - 4x + 1\): \[ h = \frac{-(-4)}{2 \cdot 2} = 1 \] Tsiva \(x = 1\) mu function: \[ k = f(1) = 2(1)^2 - 4(1) + 1 = -1 \] Saka, vertex ndiyo \((1, -1)\). 2. Akisi yeSimmetry Akisi yesimetry yeparabola mutsetse wakamira unopfuura nepa vertex:
