1428=(18+2x)(26+2x) eching

$$1428 = ( 18 + 2 x ) ( 26 + 2 x )$$

x uchun yechish

$$x=-30$$

$$x=8$$

Kvadrat tenglamalaridan foydalanish qoidalari

18+2x ga 26+2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$1428=468+88x+4x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$468+88x+4x^{2}=1428$$

Ikkala tarafdan 1428 ni ayirish.

$$468+88x+4x^{2}-1428=0$$

-960 olish uchun 468 dan 1428 ni ayirish.

$$-960+88x+4x^{2}=0$$

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.

$$4x^{2}+88x-960=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, 88 ni b va -960 ni c bilan almashtiring.

$$x=\frac{-88±\sqrt{88^{2}-4\times 4\left(-960\right)}}{2\times 4}$$

88 kvadratini chiqarish.

$$x=\frac{-88±\sqrt{7744-4\times 4\left(-960\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-88±\sqrt{7744-16\left(-960\right)}}{2\times 4}$$

-16 ni -960 marotabaga ko'paytirish.

$$x=\frac{-88±\sqrt{7744+15360}}{2\times 4}$$

7744 ni 15360 ga qo'shish.

$$x=\frac{-88±\sqrt{23104}}{2\times 4}$$

23104 ning kvadrat ildizini chiqarish.

$$x=\frac{-88±152}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{-88±152}{8}$$

x=\frac{-88±152}{8} tenglamasini yeching, bunda ± musbat. -88 ni 152 ga qo'shish.

$$x=\frac{64}{8}$$

64 ni 8 ga bo'lish.

$$x=8$$

x=\frac{-88±152}{8} tenglamasini yeching, bunda ± manfiy. -88 dan 152 ni ayirish.

$$x=-\frac{240}{8}$$

-240 ni 8 ga bo'lish.

$$x=-30$$

Tenglama yechildi.

$$x=8 x=-30$$

Kvadratni yakunlash bo'yicha qadamlar

18+2x ga 26+2x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$1428=468+88x+4x^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$468+88x+4x^{2}=1428$$

Ikkala tarafdan 468 ni ayirish.

$$88x+4x^{2}=1428-468$$

960 olish uchun 1428 dan 468 ni ayirish.

$$88x+4x^{2}=960$$

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

$$4x^{2}+88x=960$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}+88x}{4}=\frac{960}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{88}{4}x=\frac{960}{4}$$

88 ni 4 ga bo'lish.

$$x^{2}+22x=\frac{960}{4}$$

960 ni 4 ga bo'lish.

$$x^{2}+22x=240$$

22 ni bo‘lish, x shartining koeffitsienti, 2 ga 11 olish uchun. Keyin, 11 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+22x+11^{2}=240+11^{2}$$

11 kvadratini chiqarish.

$$x^{2}+22x+121=240+121$$

240 ni 121 ga qo'shish.

$$x^{2}+22x+121=361$$

x^{2}+22x+121 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+11\right)^{2}=361$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+11\right)^{2}}=\sqrt{361}$$

Qisqartirish.

$$x+11=19 x+11=-19$$

Tenglamaning ikkala tarafidan 11 ni ayirish.

$$x=8 x=-30$$