18=-1/5x^2+12x+32 eching

$$18 = - \frac { 1 } { 5 } x ^ { 2 } + 12 x + 32$$

x uchun yechish

$$x=\sqrt{970}+30\approx 61,144823005$$

$$x=30-\sqrt{970}\approx -1,144823005$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-\frac{1}{5}x^{2}+12x+32=18$$

Ikkala tarafdan 18 ni ayirish.

$$-\frac{1}{5}x^{2}+12x+32-18=0$$

14 olish uchun 32 dan 18 ni ayirish.

$$-\frac{1}{5}x^{2}+12x+14=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -\frac{1}{5} ni a, 12 ni b va 14 ni c bilan almashtiring.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-\frac{1}{5}\right)\times 14}}{2\left(-\frac{1}{5}\right)}$$

12 kvadratini chiqarish.

$$x=\frac{-12±\sqrt{144-4\left(-\frac{1}{5}\right)\times 14}}{2\left(-\frac{1}{5}\right)}$$

-4 ni -\frac{1}{5} marotabaga ko'paytirish.

$$x=\frac{-12±\sqrt{144+\frac{4}{5}\times 14}}{2\left(-\frac{1}{5}\right)}$$

\frac{4}{5} ni 14 marotabaga ko'paytirish.

$$x=\frac{-12±\sqrt{144+\frac{56}{5}}}{2\left(-\frac{1}{5}\right)}$$

144 ni \frac{56}{5} ga qo'shish.

$$x=\frac{-12±\sqrt{\frac{776}{5}}}{2\left(-\frac{1}{5}\right)}$$

\frac{776}{5} ning kvadrat ildizini chiqarish.

$$x=\frac{-12±\frac{2\sqrt{970}}{5}}{2\left(-\frac{1}{5}\right)}$$

2 ni -\frac{1}{5} marotabaga ko'paytirish.

$$x=\frac{-12±\frac{2\sqrt{970}}{5}}{-\frac{2}{5}}$$

x=\frac{-12±\frac{2\sqrt{970}}{5}}{-\frac{2}{5}} tenglamasini yeching, bunda ± musbat. -12 ni \frac{2\sqrt{970}}{5} ga qo'shish.

$$x=\frac{\frac{2\sqrt{970}}{5}-12}{-\frac{2}{5}}$$

-12+\frac{2\sqrt{970}}{5} ni -\frac{2}{5} ga bo'lish -12+\frac{2\sqrt{970}}{5} ga k'paytirish -\frac{2}{5} ga qaytarish.

$$x=30-\sqrt{970}$$

x=\frac{-12±\frac{2\sqrt{970}}{5}}{-\frac{2}{5}} tenglamasini yeching, bunda ± manfiy. -12 dan \frac{2\sqrt{970}}{5} ni ayirish.

$$x=\frac{-\frac{2\sqrt{970}}{5}-12}{-\frac{2}{5}}$$

-12-\frac{2\sqrt{970}}{5} ni -\frac{2}{5} ga bo'lish -12-\frac{2\sqrt{970}}{5} ga k'paytirish -\frac{2}{5} ga qaytarish.

$$x=\sqrt{970}+30$$

Tenglama yechildi.

$$x=30-\sqrt{970} x=\sqrt{970}+30$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$-\frac{1}{5}x^{2}+12x+32=18$$

Ikkala tarafdan 32 ni ayirish.

$$-\frac{1}{5}x^{2}+12x=18-32$$

-14 olish uchun 18 dan 32 ni ayirish.

$$-\frac{1}{5}x^{2}+12x=-14$$

Ikkala tarafini -5 ga ko‘paytiring.

$$\frac{-\frac{1}{5}x^{2}+12x}{-\frac{1}{5}}=-\frac{14}{-\frac{1}{5}}$$

-\frac{1}{5} ga bo'lish -\frac{1}{5} ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{12}{-\frac{1}{5}}x=-\frac{14}{-\frac{1}{5}}$$

12 ni -\frac{1}{5} ga bo'lish 12 ga k'paytirish -\frac{1}{5} ga qaytarish.

$$x^{2}-60x=-\frac{14}{-\frac{1}{5}}$$

-14 ni -\frac{1}{5} ga bo'lish -14 ga k'paytirish -\frac{1}{5} ga qaytarish.

$$x^{2}-60x=70$$

-60 ni bo‘lish, x shartining koeffitsienti, 2 ga -30 olish uchun. Keyin, -30 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-60x+\left(-30\right)^{2}=70+\left(-30\right)^{2}$$

-30 kvadratini chiqarish.

$$x^{2}-60x+900=70+900$$

70 ni 900 ga qo'shish.

$$x^{2}-60x+900=970$$

x^{2}-60x+900 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-30\right)^{2}=970$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-30\right)^{2}}=\sqrt{970}$$

Qisqartirish.

$$x-30=\sqrt{970} x-30=-\sqrt{970}$$

30 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\sqrt{970}+30 x=30-\sqrt{970}$$