10x^2-15x+2=0 eching

$$10 x ^ { 2 } - 15 x + 2 = 0$$

x uchun yechish

$$x=\frac{\sqrt{145}}{20}+\frac{3}{4}\approx 1,352079729$$

$$x=-\frac{\sqrt{145}}{20}+\frac{3}{4}\approx 0,147920271$$

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.

$$10x^{2}-15x+2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 10 ni a, -15 ni b va 2 ni c bilan almashtiring.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 10\times 2}}{2\times 10}$$

-15 kvadratini chiqarish.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 10\times 2}}{2\times 10}$$

-4 ni 10 marotabaga ko'paytirish.

$$x=\frac{-\left(-15\right)±\sqrt{225-40\times 2}}{2\times 10}$$

-40 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-15\right)±\sqrt{225-80}}{2\times 10}$$

225 ni -80 ga qo'shish.

$$x=\frac{-\left(-15\right)±\sqrt{145}}{2\times 10}$$

-15 ning teskarisi 15 ga teng.

$$x=\frac{15±\sqrt{145}}{2\times 10}$$

2 ni 10 marotabaga ko'paytirish.

$$x=\frac{15±\sqrt{145}}{20}$$

x=\frac{15±\sqrt{145}}{20} tenglamasini yeching, bunda ± musbat. 15 ni \sqrt{145} ga qo'shish.

$$x=\frac{\sqrt{145}+15}{20}$$

15+\sqrt{145} ni 20 ga bo'lish.

$$x=\frac{\sqrt{145}}{20}+\frac{3}{4}$$

x=\frac{15±\sqrt{145}}{20} tenglamasini yeching, bunda ± manfiy. 15 dan \sqrt{145} ni ayirish.

$$x=\frac{15-\sqrt{145}}{20}$$

15-\sqrt{145} ni 20 ga bo'lish.

$$x=-\frac{\sqrt{145}}{20}+\frac{3}{4}$$

Tenglama yechildi.

$$x=\frac{\sqrt{145}}{20}+\frac{3}{4} x=-\frac{\sqrt{145}}{20}+\frac{3}{4}$$

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.

$$10x^{2}-15x+2=0$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

$$10x^{2}-15x+2-2=-2$$

O‘zidan 2 ayirilsa 0 qoladi.

$$10x^{2}-15x=-2$$

Ikki tarafini 10 ga bo‘ling.

$$\frac{10x^{2}-15x}{10}=-\frac{2}{10}$$

10 ga bo'lish 10 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{15}{10}\right)x=-\frac{2}{10}$$

\frac{-15}{10} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{3}{2}x=-\frac{2}{10}$$

\frac{-2}{10} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{3}{2}x=-\frac{1}{5}$$

-\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{4} olish uchun. Keyin, -\frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=-\frac{1}{5}+\left(-\frac{3}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=-\frac{1}{5}+\frac{9}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{5} ni \frac{9}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{29}{80}$$

x^{2}-\frac{3}{2}x+\frac{9}{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.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{29}{80}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{29}{80}}$$

Qisqartirish.

$$x-\frac{3}{4}=\frac{\sqrt{145}}{20} x-\frac{3}{4}=-\frac{\sqrt{145}}{20}$$

\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{145}}{20}+\frac{3}{4} x=-\frac{\sqrt{145}}{20}+\frac{3}{4}$$