35*15=(19-x)(15+x) eching

$$35 \times 15=(19-x)(15+x)$$

x uchun yechish (complex solution)

$$x=2+2\sqrt{59}i\approx 2+15,362291496i$$

$$x=-2\sqrt{59}i+2\approx 2-15,362291496i$$

Kvadrat tenglamalaridan foydalanish qoidalari

525 hosil qilish uchun 35 va 15 ni ko'paytirish.

$$525=\left(19-x\right)\left(15+x\right)$$

19-x ga 15+x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$525=285+4x-x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$285+4x-x^{2}=525$$

Ikkala tarafdan 525 ni ayirish.

$$285+4x-x^{2}-525=0$$

-240 olish uchun 285 dan 525 ni ayirish.

$$-240+4x-x^{2}=0$$

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.

$$-x^{2}+4x-240=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 4 ni b va -240 ni c bilan almashtiring.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\left(-240\right)}}{2\left(-1\right)}$$

4 kvadratini chiqarish.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\left(-240\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16+4\left(-240\right)}}{2\left(-1\right)}$$

4 ni -240 marotabaga ko'paytirish.

$$x=\frac{-4±\sqrt{16-960}}{2\left(-1\right)}$$

16 ni -960 ga qo'shish.

$$x=\frac{-4±\sqrt{-944}}{2\left(-1\right)}$$

-944 ning kvadrat ildizini chiqarish.

$$x=\frac{-4±4\sqrt{59}i}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-4±4\sqrt{59}i}{-2}$$

x=\frac{-4±4\sqrt{59}i}{-2} tenglamasini yeching, bunda ± musbat. -4 ni 4i\sqrt{59} ga qo'shish.

$$x=\frac{-4+4\sqrt{59}i}{-2}$$

-4+4i\sqrt{59} ni -2 ga bo'lish.

$$x=-2\sqrt{59}i+2$$

x=\frac{-4±4\sqrt{59}i}{-2} tenglamasini yeching, bunda ± manfiy. -4 dan 4i\sqrt{59} ni ayirish.

$$x=\frac{-4\sqrt{59}i-4}{-2}$$

-4-4i\sqrt{59} ni -2 ga bo'lish.

$$x=2+2\sqrt{59}i$$

Tenglama yechildi.

$$x=-2\sqrt{59}i+2 x=2+2\sqrt{59}i$$

Kvadratni yakunlash bo'yicha qadamlar

525 hosil qilish uchun 35 va 15 ni ko'paytirish.

$$525=\left(19-x\right)\left(15+x\right)$$

19-x ga 15+x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$525=285+4x-x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$285+4x-x^{2}=525$$

Ikkala tarafdan 285 ni ayirish.

$$4x-x^{2}=525-285$$

240 olish uchun 525 dan 285 ni ayirish.

$$4x-x^{2}=240$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-x^{2}+4x=240$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+4x}{-1}=\frac{240}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{4}{-1}x=\frac{240}{-1}$$

4 ni -1 ga bo'lish.

$$x^{2}-4x=\frac{240}{-1}$$

240 ni -1 ga bo'lish.

$$x^{2}-4x=-240$$

-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.

$$x^{2}-4x+\left(-2\right)^{2}=-240+\left(-2\right)^{2}$$

-2 kvadratini chiqarish.

$$x^{2}-4x+4=-240+4$$

-240 ni 4 ga qo'shish.

$$x^{2}-4x+4=-236$$

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.

$$\left(x-2\right)^{2}=-236$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-236}$$

Qisqartirish.

$$x-2=2\sqrt{59}i x-2=-2\sqrt{59}i$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$x=2+2\sqrt{59}i x=-2\sqrt{59}i+2$$