+54x+sqrt{x}=75 eching

$$+54x+ \sqrt{ x } = 75$$

x uchun yechish

$$x = \frac{8101 - \sqrt{16201}}{5832} \approx 1,3672354$$

Yechim qadamlari

Tenglamaning ikkala tarafidan 54x ni ayirish.

$$\sqrt{x}=75-54x$$

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{x}\right)^{2}=\left(75-54x\right)^{2}$$

2 daraja ko‘rsatkichini \sqrt{x} ga hisoblang va x ni qiymatni oling.

$$x=\left(75-54x\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(75-54x\right)^{2} kengaytirilishi uchun ishlating.

$$x=5625-8100x+2916x^{2}$$

Ikkala tarafdan 5625 ni ayirish.

$$x-5625=-8100x+2916x^{2}$$

8100x ni ikki tarafga qo’shing.

$$x-5625+8100x=2916x^{2}$$

8101x ni olish uchun x va 8100x ni birlashtirish.

$$8101x-5625=2916x^{2}$$

Ikkala tarafdan 2916x^{2} ni ayirish.

$$8101x-5625-2916x^{2}=0$$

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.

$$-2916x^{2}+8101x-5625=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -2916 ni a, 8101 ni b va -5625 ni c bilan almashtiring.

$$x=\frac{-8101±\sqrt{8101^{2}-4\left(-2916\right)\left(-5625\right)}}{2\left(-2916\right)}$$

8101 kvadratini chiqarish.

$$x=\frac{-8101±\sqrt{65626201-4\left(-2916\right)\left(-5625\right)}}{2\left(-2916\right)}$$

-4 ni -2916 marotabaga ko'paytirish.

$$x=\frac{-8101±\sqrt{65626201+11664\left(-5625\right)}}{2\left(-2916\right)}$$

11664 ni -5625 marotabaga ko'paytirish.

$$x=\frac{-8101±\sqrt{65626201-65610000}}{2\left(-2916\right)}$$

65626201 ni -65610000 ga qo'shish.

$$x=\frac{-8101±\sqrt{16201}}{2\left(-2916\right)}$$

2 ni -2916 marotabaga ko'paytirish.

$$x=\frac{-8101±\sqrt{16201}}{-5832}$$

x=\frac{-8101±\sqrt{16201}}{-5832} tenglamasini yeching, bunda ± musbat. -8101 ni \sqrt{16201} ga qo'shish.

$$x=\frac{\sqrt{16201}-8101}{-5832}$$

-8101+\sqrt{16201} ni -5832 ga bo'lish.

$$x=\frac{8101-\sqrt{16201}}{5832}$$

x=\frac{-8101±\sqrt{16201}}{-5832} tenglamasini yeching, bunda ± manfiy. -8101 dan \sqrt{16201} ni ayirish.

$$x=\frac{-\sqrt{16201}-8101}{-5832}$$

-8101-\sqrt{16201} ni -5832 ga bo'lish.

$$x=\frac{\sqrt{16201}+8101}{5832}$$

Tenglama yechildi.

$$x=\frac{8101-\sqrt{16201}}{5832} x=\frac{\sqrt{16201}+8101}{5832}$$

54x+\sqrt{x}=75 tenglamasida x uchun \frac{8101-\sqrt{16201}}{5832} ni almashtiring.

$$54\times \frac{8101-\sqrt{16201}}{5832}+\sqrt{\frac{8101-\sqrt{16201}}{5832}}=75$$

Qisqartirish. x=\frac{8101-\sqrt{16201}}{5832} tenglamani qoniqtiradi.

$$75=75$$

54x+\sqrt{x}=75 tenglamasida x uchun \frac{\sqrt{16201}+8101}{5832} ni almashtiring.

$$54\times \frac{\sqrt{16201}+8101}{5832}+\sqrt{\frac{\sqrt{16201}+8101}{5832}}=75$$

Qisqartirish. x=\frac{\sqrt{16201}+8101}{5832} qiymati bu tenglamani qoniqtirmaydi.

$$\frac{1}{54}\times 16201^{\frac{1}{2}}+\frac{4051}{54}=75$$

\sqrt{x}=75-54x tenglamasi noyob yechimga ega.

$$x=\frac{8101-\sqrt{16201}}{5832}$$