98x^2+40x-30=0 eching

$$98 x ^ { 2 } + 40 x - 30 = 0$$

x uchun yechish

$$x=\frac{\sqrt{835}-10}{49}\approx 0,385640134$$

$$x=\frac{-\sqrt{835}-10}{49}\approx -0,793803399$$

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.

$$98x^{2}+40x-30=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 98 ni a, 40 ni b va -30 ni c bilan almashtiring.

$$x=\frac{-40±\sqrt{40^{2}-4\times 98\left(-30\right)}}{2\times 98}$$

40 kvadratini chiqarish.

$$x=\frac{-40±\sqrt{1600-4\times 98\left(-30\right)}}{2\times 98}$$

-4 ni 98 marotabaga ko'paytirish.

$$x=\frac{-40±\sqrt{1600-392\left(-30\right)}}{2\times 98}$$

-392 ni -30 marotabaga ko'paytirish.

$$x=\frac{-40±\sqrt{1600+11760}}{2\times 98}$$

1600 ni 11760 ga qo'shish.

$$x=\frac{-40±\sqrt{13360}}{2\times 98}$$

13360 ning kvadrat ildizini chiqarish.

$$x=\frac{-40±4\sqrt{835}}{2\times 98}$$

2 ni 98 marotabaga ko'paytirish.

$$x=\frac{-40±4\sqrt{835}}{196}$$

x=\frac{-40±4\sqrt{835}}{196} tenglamasini yeching, bunda ± musbat. -40 ni 4\sqrt{835} ga qo'shish.

$$x=\frac{4\sqrt{835}-40}{196}$$

-40+4\sqrt{835} ni 196 ga bo'lish.

$$x=\frac{\sqrt{835}-10}{49}$$

x=\frac{-40±4\sqrt{835}}{196} tenglamasini yeching, bunda ± manfiy. -40 dan 4\sqrt{835} ni ayirish.

$$x=\frac{-4\sqrt{835}-40}{196}$$

-40-4\sqrt{835} ni 196 ga bo'lish.

$$x=\frac{-\sqrt{835}-10}{49}$$

Tenglama yechildi.

$$x=\frac{\sqrt{835}-10}{49} x=\frac{-\sqrt{835}-10}{49}$$

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.

$$98x^{2}+40x-30=0$$

30 ni tenglamaning ikkala tarafiga qo'shish.

$$98x^{2}+40x-30-\left(-30\right)=-\left(-30\right)$$

O‘zidan -30 ayirilsa 0 qoladi.

$$98x^{2}+40x=-\left(-30\right)$$

0 dan -30 ni ayirish.

$$98x^{2}+40x=30$$

Ikki tarafini 98 ga bo‘ling.

$$\frac{98x^{2}+40x}{98}=\frac{30}{98}$$

98 ga bo'lish 98 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{40}{98}x=\frac{30}{98}$$

\frac{40}{98} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{20}{49}x=\frac{30}{98}$$

\frac{30}{98} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{20}{49}x=\frac{15}{49}$$

\frac{20}{49} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{10}{49} olish uchun. Keyin, \frac{10}{49} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{20}{49}x+\left(\frac{10}{49}\right)^{2}=\frac{15}{49}+\left(\frac{10}{49}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{10}{49} kvadratini chiqarish.

$$x^{2}+\frac{20}{49}x+\frac{100}{2401}=\frac{15}{49}+\frac{100}{2401}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{15}{49} ni \frac{100}{2401} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{20}{49}x+\frac{100}{2401}=\frac{835}{2401}$$

x^{2}+\frac{20}{49}x+\frac{100}{2401} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{10}{49}\right)^{2}=\frac{835}{2401}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{10}{49}\right)^{2}}=\sqrt{\frac{835}{2401}}$$

Qisqartirish.

$$x+\frac{10}{49}=\frac{\sqrt{835}}{49} x+\frac{10}{49}=-\frac{\sqrt{835}}{49}$$

Tenglamaning ikkala tarafidan \frac{10}{49} ni ayirish.

$$x=\frac{\sqrt{835}-10}{49} x=\frac{-\sqrt{835}-10}{49}$$