49x^2+2x-15=0 eching

$$49 { x }^{ 2 } +2x-15=0$$

x uchun yechish

$$x=\frac{4\sqrt{46}-1}{49}\approx 0,533251427$$

$$x=\frac{-4\sqrt{46}-1}{49}\approx -0,574067754$$

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.

$$49x^{2}+2x-15=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 49 ni a, 2 ni b va -15 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\times 49\left(-15\right)}}{2\times 49}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\times 49\left(-15\right)}}{2\times 49}$$

-4 ni 49 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-196\left(-15\right)}}{2\times 49}$$

-196 ni -15 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+2940}}{2\times 49}$$

4 ni 2940 ga qo'shish.

$$x=\frac{-2±\sqrt{2944}}{2\times 49}$$

2944 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±8\sqrt{46}}{2\times 49}$$

2 ni 49 marotabaga ko'paytirish.

$$x=\frac{-2±8\sqrt{46}}{98}$$

x=\frac{-2±8\sqrt{46}}{98} tenglamasini yeching, bunda ± musbat. -2 ni 8\sqrt{46} ga qo'shish.

$$x=\frac{8\sqrt{46}-2}{98}$$

-2+8\sqrt{46} ni 98 ga bo'lish.

$$x=\frac{4\sqrt{46}-1}{49}$$

x=\frac{-2±8\sqrt{46}}{98} tenglamasini yeching, bunda ± manfiy. -2 dan 8\sqrt{46} ni ayirish.

$$x=\frac{-8\sqrt{46}-2}{98}$$

-2-8\sqrt{46} ni 98 ga bo'lish.

$$x=\frac{-4\sqrt{46}-1}{49}$$

Tenglama yechildi.

$$x=\frac{4\sqrt{46}-1}{49} x=\frac{-4\sqrt{46}-1}{49}$$

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.

$$49x^{2}+2x-15=0$$

15 ni tenglamaning ikkala tarafiga qo'shish.

$$49x^{2}+2x-15-\left(-15\right)=-\left(-15\right)$$

O‘zidan -15 ayirilsa 0 qoladi.

$$49x^{2}+2x=-\left(-15\right)$$

0 dan -15 ni ayirish.

$$49x^{2}+2x=15$$

Ikki tarafini 49 ga bo‘ling.

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

49 ga bo'lish 49 ga ko'paytirishni bekor qiladi.

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

\frac{2}{49} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{49} olish uchun. Keyin, \frac{1}{49} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{2}{49}x+\left(\frac{1}{49}\right)^{2}=\frac{15}{49}+\left(\frac{1}{49}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{49} kvadratini chiqarish.

$$x^{2}+\frac{2}{49}x+\frac{1}{2401}=\frac{15}{49}+\frac{1}{2401}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{15}{49} ni \frac{1}{2401} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{2}{49}x+\frac{1}{2401}=\frac{736}{2401}$$

x^{2}+\frac{2}{49}x+\frac{1}{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.

$$\left(x+\frac{1}{49}\right)^{2}=\frac{736}{2401}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{49}\right)^{2}}=\sqrt{\frac{736}{2401}}$$

Qisqartirish.

$$x+\frac{1}{49}=\frac{4\sqrt{46}}{49} x+\frac{1}{49}=-\frac{4\sqrt{46}}{49}$$

Tenglamaning ikkala tarafidan \frac{1}{49} ni ayirish.

$$x=\frac{4\sqrt{46}-1}{49} x=\frac{-4\sqrt{46}-1}{49}$$