64+16q+25q^2=(8+10q)^2 eching

$$64 + 16 q + 25 q ^ { 2 } = ( 8 + 10 q ) ^ { 2 }$$

q uchun yechish

$$q = -\frac{48}{25} = -1\frac{23}{25} = -1,92$$

$$q=0$$

Faktorlash qadamlari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(8+10q\right)^{2} kengaytirilishi uchun ishlating.

$$64+16q+25q^{2}=64+160q+100q^{2}$$

Ikkala tarafdan 64 ni ayirish.

$$64+16q+25q^{2}-64=160q+100q^{2}$$

0 olish uchun 64 dan 64 ni ayirish.

$$16q+25q^{2}=160q+100q^{2}$$

Ikkala tarafdan 160q ni ayirish.

$$16q+25q^{2}-160q=100q^{2}$$

-144q ni olish uchun 16q va -160q ni birlashtirish.

$$-144q+25q^{2}=100q^{2}$$

Ikkala tarafdan 100q^{2} ni ayirish.

$$-144q+25q^{2}-100q^{2}=0$$

-75q^{2} ni olish uchun 25q^{2} va -100q^{2} ni birlashtirish.

$$-144q-75q^{2}=0$$

q omili.

$$q\left(-144-75q\right)=0$$

Tenglamani yechish uchun q=0 va -144-75q=0 ni yeching.

$$q=0 q=-\frac{48}{25}$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(8+10q\right)^{2} kengaytirilishi uchun ishlating.

$$64+16q+25q^{2}=64+160q+100q^{2}$$

Ikkala tarafdan 64 ni ayirish.

$$64+16q+25q^{2}-64=160q+100q^{2}$$

0 olish uchun 64 dan 64 ni ayirish.

$$16q+25q^{2}=160q+100q^{2}$$

Ikkala tarafdan 160q ni ayirish.

$$16q+25q^{2}-160q=100q^{2}$$

-144q ni olish uchun 16q va -160q ni birlashtirish.

$$-144q+25q^{2}=100q^{2}$$

Ikkala tarafdan 100q^{2} ni ayirish.

$$-144q+25q^{2}-100q^{2}=0$$

-75q^{2} ni olish uchun 25q^{2} va -100q^{2} ni birlashtirish.

$$-144q-75q^{2}=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-75q^{2}-144q=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -75 ni a, -144 ni b va 0 ni c bilan almashtiring.

$$q=\frac{-\left(-144\right)±\sqrt{\left(-144\right)^{2}}}{2\left(-75\right)}$$

\left(-144\right)^{2} ning kvadrat ildizini chiqarish.

$$q=\frac{-\left(-144\right)±144}{2\left(-75\right)}$$

-144 ning teskarisi 144 ga teng.

$$q=\frac{144±144}{2\left(-75\right)}$$

2 ni -75 marotabaga ko'paytirish.

$$q=\frac{144±144}{-150}$$

q=\frac{144±144}{-150} tenglamasini yeching, bunda ± musbat. 144 ni 144 ga qo'shish.

$$q=\frac{288}{-150}$$

\frac{288}{-150} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$q=-\frac{48}{25}$$

q=\frac{144±144}{-150} tenglamasini yeching, bunda ± manfiy. 144 dan 144 ni ayirish.

$$q=\frac{0}{-150}$$

0 ni -150 ga bo'lish.

$$q=0$$

Tenglama yechildi.

$$q=-\frac{48}{25} q=0$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(8+10q\right)^{2} kengaytirilishi uchun ishlating.

$$64+16q+25q^{2}=64+160q+100q^{2}$$

Ikkala tarafdan 160q ni ayirish.

$$64+16q+25q^{2}-160q=64+100q^{2}$$

-144q ni olish uchun 16q va -160q ni birlashtirish.

$$64-144q+25q^{2}=64+100q^{2}$$

Ikkala tarafdan 100q^{2} ni ayirish.

$$64-144q+25q^{2}-100q^{2}=64$$

-75q^{2} ni olish uchun 25q^{2} va -100q^{2} ni birlashtirish.

$$64-144q-75q^{2}=64$$

Ikkala tarafdan 64 ni ayirish.

$$-144q-75q^{2}=64-64$$

0 olish uchun 64 dan 64 ni ayirish.

$$-144q-75q^{2}=0$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-75q^{2}-144q=0$$

Ikki tarafini -75 ga bo‘ling.

$$\frac{-75q^{2}-144q}{-75}=\frac{0}{-75}$$

-75 ga bo'lish -75 ga ko'paytirishni bekor qiladi.

$$q^{2}+\left(-\frac{144}{-75}\right)q=\frac{0}{-75}$$

\frac{-144}{-75} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$q^{2}+\frac{48}{25}q=\frac{0}{-75}$$

0 ni -75 ga bo'lish.

$$q^{2}+\frac{48}{25}q=0$$

\frac{48}{25} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{24}{25} olish uchun. Keyin, \frac{24}{25} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$q^{2}+\frac{48}{25}q+\left(\frac{24}{25}\right)^{2}=\left(\frac{24}{25}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{24}{25} kvadratini chiqarish.

$$q^{2}+\frac{48}{25}q+\frac{576}{625}=\frac{576}{625}$$

q^{2}+\frac{48}{25}q+\frac{576}{625} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(q+\frac{24}{25}\right)^{2}=\frac{576}{625}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(q+\frac{24}{25}\right)^{2}}=\sqrt{\frac{576}{625}}$$

Qisqartirish.

$$q+\frac{24}{25}=\frac{24}{25} q+\frac{24}{25}=-\frac{24}{25}$$

Tenglamaning ikkala tarafidan \frac{24}{25} ni ayirish.

$$q=0 q=-\frac{48}{25}$$