15x^2-525x-4500=0 eching

$$15 x ^ { 2 } - 525 x - 4500 = 0$$

x uchun yechish

$$x = \frac{5 \sqrt{97} + 35}{2} \approx 42,122144504$$

$$x=\frac{35-5\sqrt{97}}{2}\approx -7,122144504$$

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.

$$15x^{2}-525x-4500=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 15 ni a, -525 ni b va -4500 ni c bilan almashtiring.

$$x=\frac{-\left(-525\right)±\sqrt{\left(-525\right)^{2}-4\times 15\left(-4500\right)}}{2\times 15}$$

-525 kvadratini chiqarish.

$$x=\frac{-\left(-525\right)±\sqrt{275625-4\times 15\left(-4500\right)}}{2\times 15}$$

-4 ni 15 marotabaga ko'paytirish.

$$x=\frac{-\left(-525\right)±\sqrt{275625-60\left(-4500\right)}}{2\times 15}$$

-60 ni -4500 marotabaga ko'paytirish.

$$x=\frac{-\left(-525\right)±\sqrt{275625+270000}}{2\times 15}$$

275625 ni 270000 ga qo'shish.

$$x=\frac{-\left(-525\right)±\sqrt{545625}}{2\times 15}$$

545625 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-525\right)±75\sqrt{97}}{2\times 15}$$

-525 ning teskarisi 525 ga teng.

$$x=\frac{525±75\sqrt{97}}{2\times 15}$$

2 ni 15 marotabaga ko'paytirish.

$$x=\frac{525±75\sqrt{97}}{30}$$

x=\frac{525±75\sqrt{97}}{30} tenglamasini yeching, bunda ± musbat. 525 ni 75\sqrt{97} ga qo'shish.

$$x=\frac{75\sqrt{97}+525}{30}$$

525+75\sqrt{97} ni 30 ga bo'lish.

$$x=\frac{5\sqrt{97}+35}{2}$$

x=\frac{525±75\sqrt{97}}{30} tenglamasini yeching, bunda ± manfiy. 525 dan 75\sqrt{97} ni ayirish.

$$x=\frac{525-75\sqrt{97}}{30}$$

525-75\sqrt{97} ni 30 ga bo'lish.

$$x=\frac{35-5\sqrt{97}}{2}$$

Tenglama yechildi.

$$x=\frac{5\sqrt{97}+35}{2} x=\frac{35-5\sqrt{97}}{2}$$

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.

$$15x^{2}-525x-4500=0$$

4500 ni tenglamaning ikkala tarafiga qo'shish.

$$15x^{2}-525x-4500-\left(-4500\right)=-\left(-4500\right)$$

O‘zidan -4500 ayirilsa 0 qoladi.

$$15x^{2}-525x=-\left(-4500\right)$$

0 dan -4500 ni ayirish.

$$15x^{2}-525x=4500$$

Ikki tarafini 15 ga bo‘ling.

$$\frac{15x^{2}-525x}{15}=\frac{4500}{15}$$

15 ga bo'lish 15 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{525}{15}\right)x=\frac{4500}{15}$$

-525 ni 15 ga bo'lish.

$$x^{2}-35x=\frac{4500}{15}$$

4500 ni 15 ga bo'lish.

$$x^{2}-35x=300$$

-35 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{35}{2} olish uchun. Keyin, -\frac{35}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-35x+\left(-\frac{35}{2}\right)^{2}=300+\left(-\frac{35}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{35}{2} kvadratini chiqarish.

$$x^{2}-35x+\frac{1225}{4}=300+\frac{1225}{4}$$

300 ni \frac{1225}{4} ga qo'shish.

$$x^{2}-35x+\frac{1225}{4}=\frac{2425}{4}$$

x^{2}-35x+\frac{1225}{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.

$$\left(x-\frac{35}{2}\right)^{2}=\frac{2425}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{35}{2}\right)^{2}}=\sqrt{\frac{2425}{4}}$$

Qisqartirish.

$$x-\frac{35}{2}=\frac{5\sqrt{97}}{2} x-\frac{35}{2}=-\frac{5\sqrt{97}}{2}$$

\frac{35}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{5\sqrt{97}+35}{2} x=\frac{35-5\sqrt{97}}{2}$$