x^2+24x-23=0 eching

$${ x }^{ 2 } +24x-23=0$$

x uchun yechish (complex solution)

$$x=\sqrt{167}-12\approx 0,922847983$$

$$x=-\left(\sqrt{167}+12\right)\approx -24,922847983$$

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.

$$x^{2}+24x-23=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 24 ni b va -23 ni c bilan almashtiring.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-23\right)}}{2}$$

24 kvadratini chiqarish.

$$x=\frac{-24±\sqrt{576-4\left(-23\right)}}{2}$$

-4 ni -23 marotabaga ko'paytirish.

$$x=\frac{-24±\sqrt{576+92}}{2}$$

576 ni 92 ga qo'shish.

$$x=\frac{-24±\sqrt{668}}{2}$$

668 ning kvadrat ildizini chiqarish.

$$x=\frac{-24±2\sqrt{167}}{2}$$

x=\frac{-24±2\sqrt{167}}{2} tenglamasini yeching, bunda ± musbat. -24 ni 2\sqrt{167} ga qo'shish.

$$x=\frac{2\sqrt{167}-24}{2}$$

-24+2\sqrt{167} ni 2 ga bo'lish.

$$x=\sqrt{167}-12$$

x=\frac{-24±2\sqrt{167}}{2} tenglamasini yeching, bunda ± manfiy. -24 dan 2\sqrt{167} ni ayirish.

$$x=\frac{-2\sqrt{167}-24}{2}$$

-24-2\sqrt{167} ni 2 ga bo'lish.

$$x=-\sqrt{167}-12$$

Tenglama yechildi.

$$x=\sqrt{167}-12 x=-\sqrt{167}-12$$

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.

$$x^{2}+24x-23=0$$

23 ni tenglamaning ikkala tarafiga qo'shish.

$$x^{2}+24x-23-\left(-23\right)=-\left(-23\right)$$

O‘zidan -23 ayirilsa 0 qoladi.

$$x^{2}+24x=-\left(-23\right)$$

0 dan -23 ni ayirish.

$$x^{2}+24x=23$$

24 ni bo‘lish, x shartining koeffitsienti, 2 ga 12 olish uchun. Keyin, 12 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+24x+12^{2}=23+12^{2}$$

12 kvadratini chiqarish.

$$x^{2}+24x+144=23+144$$

23 ni 144 ga qo'shish.

$$x^{2}+24x+144=167$$

x^{2}+24x+144 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+12\right)^{2}=167$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+12\right)^{2}}=\sqrt{167}$$

Qisqartirish.

$$x+12=\sqrt{167} x+12=-\sqrt{167}$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$x=\sqrt{167}-12 x=-\sqrt{167}-12$$

x uchun yechish

$$x=\sqrt{167}-12\approx 0,922847983$$

$$x=-\sqrt{167}-12\approx -24,922847983$$

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.

$$x^{2}+24x-23=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 24 ni b va -23 ni c bilan almashtiring.

$$x=\frac{-24±\sqrt{24^{2}-4\left(-23\right)}}{2}$$

24 kvadratini chiqarish.

$$x=\frac{-24±\sqrt{576-4\left(-23\right)}}{2}$$

-4 ni -23 marotabaga ko'paytirish.

$$x=\frac{-24±\sqrt{576+92}}{2}$$

576 ni 92 ga qo'shish.

$$x=\frac{-24±\sqrt{668}}{2}$$

668 ning kvadrat ildizini chiqarish.

$$x=\frac{-24±2\sqrt{167}}{2}$$

x=\frac{-24±2\sqrt{167}}{2} tenglamasini yeching, bunda ± musbat. -24 ni 2\sqrt{167} ga qo'shish.

$$x=\frac{2\sqrt{167}-24}{2}$$

-24+2\sqrt{167} ni 2 ga bo'lish.

$$x=\sqrt{167}-12$$

x=\frac{-24±2\sqrt{167}}{2} tenglamasini yeching, bunda ± manfiy. -24 dan 2\sqrt{167} ni ayirish.

$$x=\frac{-2\sqrt{167}-24}{2}$$

-24-2\sqrt{167} ni 2 ga bo'lish.

$$x=-\sqrt{167}-12$$

Tenglama yechildi.

$$x=\sqrt{167}-12 x=-\sqrt{167}-12$$

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.

$$x^{2}+24x-23=0$$

23 ni tenglamaning ikkala tarafiga qo'shish.

$$x^{2}+24x-23-\left(-23\right)=-\left(-23\right)$$

O‘zidan -23 ayirilsa 0 qoladi.

$$x^{2}+24x=-\left(-23\right)$$

0 dan -23 ni ayirish.

$$x^{2}+24x=23$$

24 ni bo‘lish, x shartining koeffitsienti, 2 ga 12 olish uchun. Keyin, 12 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+24x+12^{2}=23+12^{2}$$

12 kvadratini chiqarish.

$$x^{2}+24x+144=23+144$$

23 ni 144 ga qo'shish.

$$x^{2}+24x+144=167$$

x^{2}+24x+144 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+12\right)^{2}=167$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+12\right)^{2}}=\sqrt{167}$$

Qisqartirish.

$$x+12=\sqrt{167} x+12=-\sqrt{167}$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$x=\sqrt{167}-12 x=-\sqrt{167}-12$$