x^2-13x-42=0 eching

$${ x }^{ 2 } -13x-42=0$$

x uchun yechish

$$x = \frac{\sqrt{337} + 13}{2} \approx 15,678779875$$

$$x=\frac{13-\sqrt{337}}{2}\approx -2,678779875$$

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.

$$x^{2}-13x-42=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, -13 ni b va -42 ni c bilan almashtiring.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\left(-42\right)}}{2}$$

-13 kvadratini chiqarish.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\left(-42\right)}}{2}$$

-4 ni -42 marotabaga ko'paytirish.

$$x=\frac{-\left(-13\right)±\sqrt{169+168}}{2}$$

169 ni 168 ga qo'shish.

$$x=\frac{-\left(-13\right)±\sqrt{337}}{2}$$

-13 ning teskarisi 13 ga teng.

$$x=\frac{13±\sqrt{337}}{2}$$

x=\frac{13±\sqrt{337}}{2} tenglamasini yeching, bunda ± musbat. 13 ni \sqrt{337} ga qo'shish.

$$x=\frac{\sqrt{337}+13}{2}$$

x=\frac{13±\sqrt{337}}{2} tenglamasini yeching, bunda ± manfiy. 13 dan \sqrt{337} ni ayirish.

$$x=\frac{13-\sqrt{337}}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{337}+13}{2} x=\frac{13-\sqrt{337}}{2}$$

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.

$$x^{2}-13x-42=0$$

42 ni tenglamaning ikkala tarafiga qo'shish.

$$x^{2}-13x-42-\left(-42\right)=-\left(-42\right)$$

O‘zidan -42 ayirilsa 0 qoladi.

$$x^{2}-13x=-\left(-42\right)$$

0 dan -42 ni ayirish.

$$x^{2}-13x=42$$

-13 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{13}{2} olish uchun. Keyin, -\frac{13}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=42+\left(-\frac{13}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{13}{2} kvadratini chiqarish.

$$x^{2}-13x+\frac{169}{4}=42+\frac{169}{4}$$

42 ni \frac{169}{4} ga qo'shish.

$$x^{2}-13x+\frac{169}{4}=\frac{337}{4}$$

x^{2}-13x+\frac{169}{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.

$$\left(x-\frac{13}{2}\right)^{2}=\frac{337}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{337}{4}}$$

Qisqartirish.

$$x-\frac{13}{2}=\frac{\sqrt{337}}{2} x-\frac{13}{2}=-\frac{\sqrt{337}}{2}$$

\frac{13}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{337}+13}{2} x=\frac{13-\sqrt{337}}{2}$$