4x^2-14x=9 eching

$$4 x ^ { 2 } - 14 x = 9$$

x uchun yechish

$$x = \frac{\sqrt{85} + 7}{4} \approx 4,054886114$$

$$x=\frac{7-\sqrt{85}}{4}\approx -0,554886114$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$4x^{2}-14x=9$$

Tenglamaning ikkala tarafidan 9 ni ayirish.

$$4x^{2}-14x-9=9-9$$

O‘zidan 9 ayirilsa 0 qoladi.

$$4x^{2}-14x-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -14 ni b va -9 ni c bilan almashtiring.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 4\left(-9\right)}}{2\times 4}$$

-14 kvadratini chiqarish.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 4\left(-9\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196-16\left(-9\right)}}{2\times 4}$$

-16 ni -9 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196+144}}{2\times 4}$$

196 ni 144 ga qo'shish.

$$x=\frac{-\left(-14\right)±\sqrt{340}}{2\times 4}$$

340 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-14\right)±2\sqrt{85}}{2\times 4}$$

-14 ning teskarisi 14 ga teng.

$$x=\frac{14±2\sqrt{85}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{14±2\sqrt{85}}{8}$$

x=\frac{14±2\sqrt{85}}{8} tenglamasini yeching, bunda ± musbat. 14 ni 2\sqrt{85} ga qo'shish.

$$x=\frac{2\sqrt{85}+14}{8}$$

14+2\sqrt{85} ni 8 ga bo'lish.

$$x=\frac{\sqrt{85}+7}{4}$$

x=\frac{14±2\sqrt{85}}{8} tenglamasini yeching, bunda ± manfiy. 14 dan 2\sqrt{85} ni ayirish.

$$x=\frac{14-2\sqrt{85}}{8}$$

14-2\sqrt{85} ni 8 ga bo'lish.

$$x=\frac{7-\sqrt{85}}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{85}+7}{4} x=\frac{7-\sqrt{85}}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$4x^{2}-14x=9$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-14x}{4}=\frac{9}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{14}{4}\right)x=\frac{9}{4}$$

\frac{-14}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{7}{2}x=\frac{9}{4}$$

-\frac{7}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{4} olish uchun. Keyin, -\frac{7}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=\frac{9}{4}+\left(-\frac{7}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{4} kvadratini chiqarish.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{9}{4}+\frac{49}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{4} ni \frac{49}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{85}{16}$$

x^{2}-\frac{7}{2}x+\frac{49}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{85}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{85}{16}}$$

Qisqartirish.

$$x-\frac{7}{4}=\frac{\sqrt{85}}{4} x-\frac{7}{4}=-\frac{\sqrt{85}}{4}$$

\frac{7}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{85}+7}{4} x=\frac{7-\sqrt{85}}{4}$$