x^2+20x=45 eching

$$x ^ { 2 } + 20 x = 45$$

x uchun yechish (complex solution)

$$x=\sqrt{145}-10\approx 2,041594579$$

$$x=-\left(\sqrt{145}+10\right)\approx -22,041594579$$

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.

$$x^{2}+20x=45$$

Tenglamaning ikkala tarafidan 45 ni ayirish.

$$x^{2}+20x-45=45-45$$

O‘zidan 45 ayirilsa 0 qoladi.

$$x^{2}+20x-45=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 20 ni b va -45 ni c bilan almashtiring.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-45\right)}}{2}$$

20 kvadratini chiqarish.

$$x=\frac{-20±\sqrt{400-4\left(-45\right)}}{2}$$

-4 ni -45 marotabaga ko'paytirish.

$$x=\frac{-20±\sqrt{400+180}}{2}$$

400 ni 180 ga qo'shish.

$$x=\frac{-20±\sqrt{580}}{2}$$

580 ning kvadrat ildizini chiqarish.

$$x=\frac{-20±2\sqrt{145}}{2}$$

x=\frac{-20±2\sqrt{145}}{2} tenglamasini yeching, bunda ± musbat. -20 ni 2\sqrt{145} ga qo'shish.

$$x=\frac{2\sqrt{145}-20}{2}$$

-20+2\sqrt{145} ni 2 ga bo'lish.

$$x=\sqrt{145}-10$$

x=\frac{-20±2\sqrt{145}}{2} tenglamasini yeching, bunda ± manfiy. -20 dan 2\sqrt{145} ni ayirish.

$$x=\frac{-2\sqrt{145}-20}{2}$$

-20-2\sqrt{145} ni 2 ga bo'lish.

$$x=-\sqrt{145}-10$$

Tenglama yechildi.

$$x=\sqrt{145}-10 x=-\sqrt{145}-10$$

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.

$$x^{2}+20x=45$$

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

$$x^{2}+20x+10^{2}=45+10^{2}$$

10 kvadratini chiqarish.

$$x^{2}+20x+100=45+100$$

45 ni 100 ga qo'shish.

$$x^{2}+20x+100=145$$

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

$$\left(x+10\right)^{2}=145$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{145}$$

Qisqartirish.

$$x+10=\sqrt{145} x+10=-\sqrt{145}$$

Tenglamaning ikkala tarafidan 10 ni ayirish.

$$x=\sqrt{145}-10 x=-\sqrt{145}-10$$

x uchun yechish

$$x=\sqrt{145}-10\approx 2,041594579$$

$$x=-\sqrt{145}-10\approx -22,041594579$$

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.

$$x^{2}+20x=45$$

Tenglamaning ikkala tarafidan 45 ni ayirish.

$$x^{2}+20x-45=45-45$$

O‘zidan 45 ayirilsa 0 qoladi.

$$x^{2}+20x-45=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 20 ni b va -45 ni c bilan almashtiring.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-45\right)}}{2}$$

20 kvadratini chiqarish.

$$x=\frac{-20±\sqrt{400-4\left(-45\right)}}{2}$$

-4 ni -45 marotabaga ko'paytirish.

$$x=\frac{-20±\sqrt{400+180}}{2}$$

400 ni 180 ga qo'shish.

$$x=\frac{-20±\sqrt{580}}{2}$$

580 ning kvadrat ildizini chiqarish.

$$x=\frac{-20±2\sqrt{145}}{2}$$

x=\frac{-20±2\sqrt{145}}{2} tenglamasini yeching, bunda ± musbat. -20 ni 2\sqrt{145} ga qo'shish.

$$x=\frac{2\sqrt{145}-20}{2}$$

-20+2\sqrt{145} ni 2 ga bo'lish.

$$x=\sqrt{145}-10$$

x=\frac{-20±2\sqrt{145}}{2} tenglamasini yeching, bunda ± manfiy. -20 dan 2\sqrt{145} ni ayirish.

$$x=\frac{-2\sqrt{145}-20}{2}$$

-20-2\sqrt{145} ni 2 ga bo'lish.

$$x=-\sqrt{145}-10$$

Tenglama yechildi.

$$x=\sqrt{145}-10 x=-\sqrt{145}-10$$

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.

$$x^{2}+20x=45$$

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

$$x^{2}+20x+10^{2}=45+10^{2}$$

10 kvadratini chiqarish.

$$x^{2}+20x+100=45+100$$

45 ni 100 ga qo'shish.

$$x^{2}+20x+100=145$$

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

$$\left(x+10\right)^{2}=145$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+10\right)^{2}}=\sqrt{145}$$

Qisqartirish.

$$x+10=\sqrt{145} x+10=-\sqrt{145}$$

Tenglamaning ikkala tarafidan 10 ni ayirish.

$$x=\sqrt{145}-10 x=-\sqrt{145}-10$$