2x^2-11x=2 eching

$$2 x ^ { 2 } - 11 x = 2$$

x uchun yechish

$$x = \frac{\sqrt{137} + 11}{4} \approx 5,676174978$$

$$x=\frac{11-\sqrt{137}}{4}\approx -0,176174978$$

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.

$$2x^{2}-11x=2$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$2x^{2}-11x-2=2-2$$

O‘zidan 2 ayirilsa 0 qoladi.

$$2x^{2}-11x-2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -11 ni b va -2 ni c bilan almashtiring.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 2\left(-2\right)}}{2\times 2}$$

-11 kvadratini chiqarish.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 2\left(-2\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121-8\left(-2\right)}}{2\times 2}$$

-8 ni -2 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121+16}}{2\times 2}$$

121 ni 16 ga qo'shish.

$$x=\frac{-\left(-11\right)±\sqrt{137}}{2\times 2}$$

-11 ning teskarisi 11 ga teng.

$$x=\frac{11±\sqrt{137}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{11±\sqrt{137}}{4}$$

x=\frac{11±\sqrt{137}}{4} tenglamasini yeching, bunda ± musbat. 11 ni \sqrt{137} ga qo'shish.

$$x=\frac{\sqrt{137}+11}{4}$$

x=\frac{11±\sqrt{137}}{4} tenglamasini yeching, bunda ± manfiy. 11 dan \sqrt{137} ni ayirish.

$$x=\frac{11-\sqrt{137}}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{137}+11}{4} x=\frac{11-\sqrt{137}}{4}$$

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.

$$2x^{2}-11x=2$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-11x}{2}=\frac{2}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{11}{2}x=\frac{2}{2}$$

2 ni 2 ga bo'lish.

$$x^{2}-\frac{11}{2}x=1$$

-\frac{11}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{4} olish uchun. Keyin, -\frac{11}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{11}{2}x+\left(-\frac{11}{4}\right)^{2}=1+\left(-\frac{11}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{4} kvadratini chiqarish.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=1+\frac{121}{16}$$

1 ni \frac{121}{16} ga qo'shish.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=\frac{137}{16}$$

x^{2}-\frac{11}{2}x+\frac{121}{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.

$$\left(x-\frac{11}{4}\right)^{2}=\frac{137}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{4}\right)^{2}}=\sqrt{\frac{137}{16}}$$

Qisqartirish.

$$x-\frac{11}{4}=\frac{\sqrt{137}}{4} x-\frac{11}{4}=-\frac{\sqrt{137}}{4}$$

\frac{11}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{137}+11}{4} x=\frac{11-\sqrt{137}}{4}$$