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Ufumbuzi - Kutatua equations za quadratic kwa kukamilisha mraba

Fomu halisi:

x_{1} = \frac{5}{8} + \frac{\sqrt{41}}{8}

x_1=\frac{5}{8}+\frac{\sqrt{41}}{8}

x_{2} = \frac{5}{8} - \frac{\sqrt{41}}{8}

x_2=\frac{5}{8}-\frac{\sqrt{41}}{8}

Fomu ya desimali:

x_{1} = 1.425

x_1=1.425

x_{2} = - 0.175

x_2=-0.175

Tazama hatua

Maelezo kwa hatua

1. Tambua coefficients

Tumia fomu standard ya equation ya quadratic, $a x^{2} + b x + c = 0$ , ili kupata coefficients:

$4 x^{2} - 5 x - 1 = 0$

$a = 4$
$b = - 5$
$c = - 1$

2. Fanya coefficient a iwe sawa na 1

Kwa sababu $a = 4$ , gawa coefficients zote na constants kwa pande zote za equation na $4$ :

$4 x^{2} - 5 x - 1 = 0$

$\frac{4}{4} x^{2} - 5 \frac{x}{4} - \frac{1}{4} = \frac{0}{4}$

Rahisisha usemi

$x^{2} - \frac{5}{4} x - \frac{1}{4} = 0$

Vigawanyaji ni:
$a = 1$
$b = - \frac{5}{4}$
$c = - \frac{1}{4}$

3. Hamisha constant upande wa kulia wa equation na unganisha

Ongeza $\frac{1}{4}$ kwa pande zote za equation:

$x^{2} - \frac{5}{4} x - \frac{1}{4} = 0$

$x^{2} - \frac{5}{4} x - \frac{1}{4} + \frac{1}{4} = 0 + \frac{1}{4}$

$x^{2} - \frac{5}{4} x = \frac{1}{4}$

4. Kamilisha square

Ili kufanya upande wa kushoto wa equation kuwa perfect square trinomial, ongeza constant mpya sawa na ${(\frac{b}{2})}^{2}$ kwa equation:

$b = - \frac{5}{4}$

${(\frac{b}{2})}^{2} = {(\frac{- \frac{5}{4}}{2})}^{2}$

Tumia kanuni ya kipeo cha sehemu ${(\frac{x}{y})}^{2} = \frac{x^{2}}{y^{2}}$

${(\frac{- \frac{5}{4}}{2})}^{2} = \frac{{(- \frac{5}{4})}^{2}}{2^{2}}$

$\frac{{(- \frac{5}{4})}^{2}}{2^{2}} = \frac{\frac{25}{16}}{4}$

$\frac{\frac{25}{16}}{4} = \frac{25}{16} \cdot \frac{1}{4}$

$\frac{25}{16} \cdot \frac{1}{4} = \frac{25}{64}$

Ongeza $\frac{25}{64}$ kwa pande zote za equation:

5 ziada steps

$x^{2} - \frac{5}{4} x = \frac{1}{4}$

$x^{2} - \frac{5}{4} x + \frac{25}{64} = \frac{1}{4} + \frac{25}{64}$

Pata msingi wa kawaida mdogo:

$x^{2} - \frac{5}{4} x + \frac{25}{64} = \frac{(1 \cdot 16)}{(4 \cdot 16)} + \frac{25}{64}$

Zidisha misingi:

$x^{2} - \frac{5}{4} x + \frac{25}{64} = \frac{(1 \cdot 16)}{64} + \frac{25}{64}$

Zidisha namba za juu:

$x^{2} - \frac{5}{4} x + \frac{25}{64} = \frac{16}{64} + \frac{25}{64}$

Unganisha sehemu:

$x^{2} - \frac{5}{4} x + \frac{25}{64} = \frac{(16 + 25)}{64}$

Unganisha namba za juu:

$x^{2} - \frac{5}{4} x + \frac{25}{64} = \frac{41}{64}$

Sasa tuna perfect square trinomial, tunaweza kuandika kama fomu ya perfect square kwa kuongezea nusu ya coefficient ya $b$ , $\frac{b}{2}$ :
$b = - \frac{5}{4}$

2 ziada steps

$\frac{b}{2} = \frac{- \frac{5}{4}}{2}$

Rahisisha kugawa:

$\frac{b}{2} = \frac{- 5}{(4 \cdot 2)}$

Rahisisha hesabu:

$\frac{b}{2} = \frac{- 5}{8}$

$x^{2} - \frac{5}{4} x + \frac{25}{64} = \frac{41}{64}$

${(x - \frac{5}{8})}^{2} = \frac{41}{64}$

5. Suluhisha kwa $x$

Chukua square root ya pande zote za equation: IMPORTANT: Wakati wa kutafuta square root ya constant, tunapata solutions mbili: positive na negative

${(x - \frac{5}{8})}^{2} = \frac{41}{64}$

$\sqrt{{(x - \frac{5}{8})}^{2}} = \sqrt{\frac{41}{64}}$

Futa mraba na mzizi mraba kwa upande wa kushoto wa equation:

$x - \frac{5}{8} = \pm \sqrt{\frac{41}{64}}$

Jumlisha $\frac{5}{8}$ kwa pande zote mbili

$x - \frac{5}{8} + \frac{5}{8} = \frac{5}{8} \pm \sqrt{\frac{41}{64}}$

Rahisisha upande wa kushoto:

$x = \frac{5}{8} \pm \sqrt{\frac{41}{64}}$

$x = \frac{5}{8} \pm \frac{\sqrt{41}}{\sqrt{64}}$

$x = \frac{5}{8} \pm \frac{\sqrt{41}}{8}$

$x_{1} = \frac{5}{8} + \frac{\sqrt{41}}{8}$
$x_{2} = \frac{5}{8} - \frac{\sqrt{41}}{8}$

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Kwa nini kujifunza hii

Katika kazi yao msingi zaidi, equation mraba zinafafanua umbo kama duru, elipisi na parabolas. Hizi maumbo zinaweza kutumika kutabiri njia ya kitu katika harakati, kama mpira uliyopigwa na mchezaji wa mpira wa miguu au uliyotupwa kutoka kwenye kanyonya.
Linapokuja suala la harakati ya kitu kwenye anga, mahali pazuri pa kuanza ni anga yenyewe, na mapinduzi ya sayari kuzunguka jua katika mfumo wetu wa jua. Equation mraba ilitumika kuanzisha kuwa mizunguko ya sayari ni elliptical, si ya mduara. Kutambua njia na speed kitu safiri kwenye anga inawezekana hata baada ya kitu kuja kuacha: the equation mraba inaweza kuhesabu speed gari ilikuwa inaendesha wakati ilipata ajali. Kwa informesheni kama hii, sekta ya magari inaweza kubuni breki kuzuia ajali kwa siku za usoni. Viwanda vingi vinatumia equation mraba kutabiri na hivyo kuboresha muda wa maisha ya bidhaa zao na usalama.

Vigezo na mada

Kutatua milinganyo ya quadratiki kwa kukamilisha mraba