sqrt{x}=5x+3 eching

$$\sqrt{ x } =5x+3$$

x uchun yechish (complex solution)

$$x=\frac{-29+\sqrt{59}i}{50}\approx -0,58+0,153622915i$$

Yechim qadamlari

Tenglamaning ikkala taraf kvadratini chiqarish.

$$\left(\sqrt{x}\right)^{2}=\left(5x+3\right)^{2}$$

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

$$x=\left(5x+3\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(5x+3\right)^{2} kengaytirilishi uchun ishlating.

$$x=25x^{2}+30x+9$$

Ikkala tarafdan 25x^{2} ni ayirish.

$$x-25x^{2}=30x+9$$

Ikkala tarafdan 30x ni ayirish.

$$x-25x^{2}-30x=9$$

-29x ni olish uchun x va -30x ni birlashtirish.

$$-29x-25x^{2}=9$$

Ikkala tarafdan 9 ni ayirish.

$$-29x-25x^{2}-9=0$$

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.

$$-25x^{2}-29x-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -25 ni a, -29 ni b va -9 ni c bilan almashtiring.

$$x=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\left(-25\right)\left(-9\right)}}{2\left(-25\right)}$$

-29 kvadratini chiqarish.

$$x=\frac{-\left(-29\right)±\sqrt{841-4\left(-25\right)\left(-9\right)}}{2\left(-25\right)}$$

-4 ni -25 marotabaga ko'paytirish.

$$x=\frac{-\left(-29\right)±\sqrt{841+100\left(-9\right)}}{2\left(-25\right)}$$

100 ni -9 marotabaga ko'paytirish.

$$x=\frac{-\left(-29\right)±\sqrt{841-900}}{2\left(-25\right)}$$

841 ni -900 ga qo'shish.

$$x=\frac{-\left(-29\right)±\sqrt{-59}}{2\left(-25\right)}$$

-59 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-29\right)±\sqrt{59}i}{2\left(-25\right)}$$

-29 ning teskarisi 29 ga teng.

$$x=\frac{29±\sqrt{59}i}{2\left(-25\right)}$$

2 ni -25 marotabaga ko'paytirish.

$$x=\frac{29±\sqrt{59}i}{-50}$$

x=\frac{29±\sqrt{59}i}{-50} tenglamasini yeching, bunda ± musbat. 29 ni i\sqrt{59} ga qo'shish.

$$x=\frac{29+\sqrt{59}i}{-50}$$

29+i\sqrt{59} ni -50 ga bo'lish.

$$x=\frac{-\sqrt{59}i-29}{50}$$

x=\frac{29±\sqrt{59}i}{-50} tenglamasini yeching, bunda ± manfiy. 29 dan i\sqrt{59} ni ayirish.

$$x=\frac{-\sqrt{59}i+29}{-50}$$

29-i\sqrt{59} ni -50 ga bo'lish.

$$x=\frac{-29+\sqrt{59}i}{50}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{59}i-29}{50} x=\frac{-29+\sqrt{59}i}{50}$$

\sqrt{x}=5x+3 tenglamasida x uchun \frac{-\sqrt{59}i-29}{50} ni almashtiring.

$$\sqrt{\frac{-\sqrt{59}i-29}{50}}=5\times \frac{-\sqrt{59}i-29}{50}+3$$

Qisqartirish. x=\frac{-\sqrt{59}i-29}{50} qiymati bu tenglamani qoniqtirmaydi.

$$-\left(\frac{1}{10}-\frac{1}{10}i\times 59^{\frac{1}{2}}\right)=-\frac{1}{10}i\times 59^{\frac{1}{2}}+\frac{1}{10}$$

\sqrt{x}=5x+3 tenglamasida x uchun \frac{-29+\sqrt{59}i}{50} ni almashtiring.

$$\sqrt{\frac{-29+\sqrt{59}i}{50}}=5\times \frac{-29+\sqrt{59}i}{50}+3$$

Qisqartirish. x=\frac{-29+\sqrt{59}i}{50} tenglamani qoniqtiradi.

$$\frac{1}{10}+\frac{1}{10}i\times 59^{\frac{1}{2}}=\frac{1}{10}+\frac{1}{10}i\times 59^{\frac{1}{2}}$$

\sqrt{x}=5x+3 tenglamasi noyob yechimga ega.

$$x=\frac{-29+\sqrt{59}i}{50}$$