-16x^2+64x-60=60 eching

$$-16 { x }^{ 2 } +64x-60=60$$

x uchun yechish (complex solution)

$$x=\frac{\sqrt{14}i}{2}+2\approx 2+1,870828693i$$

$$x=-\frac{\sqrt{14}i}{2}+2\approx 2-1,870828693i$$

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.

$$-16x^{2}+64x-60=60$$

Tenglamaning ikkala tarafidan 60 ni ayirish.

$$-16x^{2}+64x-60-60=60-60$$

O‘zidan 60 ayirilsa 0 qoladi.

$$-16x^{2}+64x-60-60=0$$

-60 dan 60 ni ayirish.

$$-16x^{2}+64x-120=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -16 ni a, 64 ni b va -120 ni c bilan almashtiring.

$$x=\frac{-64±\sqrt{64^{2}-4\left(-16\right)\left(-120\right)}}{2\left(-16\right)}$$

64 kvadratini chiqarish.

$$x=\frac{-64±\sqrt{4096-4\left(-16\right)\left(-120\right)}}{2\left(-16\right)}$$

-4 ni -16 marotabaga ko'paytirish.

$$x=\frac{-64±\sqrt{4096+64\left(-120\right)}}{2\left(-16\right)}$$

64 ni -120 marotabaga ko'paytirish.

$$x=\frac{-64±\sqrt{4096-7680}}{2\left(-16\right)}$$

4096 ni -7680 ga qo'shish.

$$x=\frac{-64±\sqrt{-3584}}{2\left(-16\right)}$$

-3584 ning kvadrat ildizini chiqarish.

$$x=\frac{-64±16\sqrt{14}i}{2\left(-16\right)}$$

2 ni -16 marotabaga ko'paytirish.

$$x=\frac{-64±16\sqrt{14}i}{-32}$$

x=\frac{-64±16\sqrt{14}i}{-32} tenglamasini yeching, bunda ± musbat. -64 ni 16i\sqrt{14} ga qo'shish.

$$x=\frac{-64+16\sqrt{14}i}{-32}$$

-64+16i\sqrt{14} ni -32 ga bo'lish.

$$x=-\frac{\sqrt{14}i}{2}+2$$

x=\frac{-64±16\sqrt{14}i}{-32} tenglamasini yeching, bunda ± manfiy. -64 dan 16i\sqrt{14} ni ayirish.

$$x=\frac{-16\sqrt{14}i-64}{-32}$$

-64-16i\sqrt{14} ni -32 ga bo'lish.

$$x=\frac{\sqrt{14}i}{2}+2$$

Tenglama yechildi.

$$x=-\frac{\sqrt{14}i}{2}+2 x=\frac{\sqrt{14}i}{2}+2$$

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.

$$-16x^{2}+64x-60=60$$

60 ni tenglamaning ikkala tarafiga qo'shish.

$$-16x^{2}+64x-60-\left(-60\right)=60-\left(-60\right)$$

O‘zidan -60 ayirilsa 0 qoladi.

$$-16x^{2}+64x=60-\left(-60\right)$$

60 dan -60 ni ayirish.

$$-16x^{2}+64x=120$$

Ikki tarafini -16 ga bo‘ling.

$$\frac{-16x^{2}+64x}{-16}=\frac{120}{-16}$$

-16 ga bo'lish -16 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{64}{-16}x=\frac{120}{-16}$$

64 ni -16 ga bo'lish.

$$x^{2}-4x=\frac{120}{-16}$$

\frac{120}{-16} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-4x=-\frac{15}{2}$$

-4 ni bo‘lish, x shartining koeffitsienti, 2 ga -2 olish uchun. Keyin, -2 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-4x+\left(-2\right)^{2}=-\frac{15}{2}+\left(-2\right)^{2}$$

-2 kvadratini chiqarish.

$$x^{2}-4x+4=-\frac{15}{2}+4$$

-\frac{15}{2} ni 4 ga qo'shish.

$$x^{2}-4x+4=-\frac{7}{2}$$

x^{2}-4x+4 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-2\right)^{2}=-\frac{7}{2}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-\frac{7}{2}}$$

Qisqartirish.

$$x-2=\frac{\sqrt{14}i}{2} x-2=-\frac{\sqrt{14}i}{2}$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{14}i}{2}+2 x=-\frac{\sqrt{14}i}{2}+2$$