0=y^2+6y-14 eching

$$0 = y ^ { 2 } + 6 y - 14$$

y uchun yechish (complex solution)

$$y=\sqrt{23}-3\approx 1,795831523$$

$$y=-\left(\sqrt{23}+3\right)\approx -7,795831523$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$y^{2}+6y-14=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 6 ni b va -14 ni c bilan almashtiring.

$$y=\frac{-6±\sqrt{6^{2}-4\left(-14\right)}}{2}$$

6 kvadratini chiqarish.

$$y=\frac{-6±\sqrt{36-4\left(-14\right)}}{2}$$

-4 ni -14 marotabaga ko'paytirish.

$$y=\frac{-6±\sqrt{36+56}}{2}$$

36 ni 56 ga qo'shish.

$$y=\frac{-6±\sqrt{92}}{2}$$

92 ning kvadrat ildizini chiqarish.

$$y=\frac{-6±2\sqrt{23}}{2}$$

y=\frac{-6±2\sqrt{23}}{2} tenglamasini yeching, bunda ± musbat. -6 ni 2\sqrt{23} ga qo'shish.

$$y=\frac{2\sqrt{23}-6}{2}$$

-6+2\sqrt{23} ni 2 ga bo'lish.

$$y=\sqrt{23}-3$$

y=\frac{-6±2\sqrt{23}}{2} tenglamasini yeching, bunda ± manfiy. -6 dan 2\sqrt{23} ni ayirish.

$$y=\frac{-2\sqrt{23}-6}{2}$$

-6-2\sqrt{23} ni 2 ga bo'lish.

$$y=-\sqrt{23}-3$$

Tenglama yechildi.

$$y=\sqrt{23}-3 y=-\sqrt{23}-3$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$y^{2}+6y-14=0$$

14 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$y^{2}+6y=14$$

6 ni bo‘lish, x shartining koeffitsienti, 2 ga 3 olish uchun. Keyin, 3 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$y^{2}+6y+3^{2}=14+3^{2}$$

3 kvadratini chiqarish.

$$y^{2}+6y+9=14+9$$

14 ni 9 ga qo'shish.

$$y^{2}+6y+9=23$$

y^{2}+6y+9 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(y+3\right)^{2}=23$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y+3\right)^{2}}=\sqrt{23}$$

Qisqartirish.

$$y+3=\sqrt{23} y+3=-\sqrt{23}$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$y=\sqrt{23}-3 y=-\sqrt{23}-3$$

y uchun yechish

$$y=\sqrt{23}-3\approx 1,795831523$$

$$y=-\sqrt{23}-3\approx -7,795831523$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$y^{2}+6y-14=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1 ni a, 6 ni b va -14 ni c bilan almashtiring.

$$y=\frac{-6±\sqrt{6^{2}-4\left(-14\right)}}{2}$$

6 kvadratini chiqarish.

$$y=\frac{-6±\sqrt{36-4\left(-14\right)}}{2}$$

-4 ni -14 marotabaga ko'paytirish.

$$y=\frac{-6±\sqrt{36+56}}{2}$$

36 ni 56 ga qo'shish.

$$y=\frac{-6±\sqrt{92}}{2}$$

92 ning kvadrat ildizini chiqarish.

$$y=\frac{-6±2\sqrt{23}}{2}$$

y=\frac{-6±2\sqrt{23}}{2} tenglamasini yeching, bunda ± musbat. -6 ni 2\sqrt{23} ga qo'shish.

$$y=\frac{2\sqrt{23}-6}{2}$$

-6+2\sqrt{23} ni 2 ga bo'lish.

$$y=\sqrt{23}-3$$

y=\frac{-6±2\sqrt{23}}{2} tenglamasini yeching, bunda ± manfiy. -6 dan 2\sqrt{23} ni ayirish.

$$y=\frac{-2\sqrt{23}-6}{2}$$

-6-2\sqrt{23} ni 2 ga bo'lish.

$$y=-\sqrt{23}-3$$

Tenglama yechildi.

$$y=\sqrt{23}-3 y=-\sqrt{23}-3$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$y^{2}+6y-14=0$$

14 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$y^{2}+6y=14$$

6 ni bo‘lish, x shartining koeffitsienti, 2 ga 3 olish uchun. Keyin, 3 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$y^{2}+6y+3^{2}=14+3^{2}$$

3 kvadratini chiqarish.

$$y^{2}+6y+9=14+9$$

14 ni 9 ga qo'shish.

$$y^{2}+6y+9=23$$

y^{2}+6y+9 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(y+3\right)^{2}=23$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(y+3\right)^{2}}=\sqrt{23}$$

Qisqartirish.

$$y+3=\sqrt{23} y+3=-\sqrt{23}$$

Tenglamaning ikkala tarafidan 3 ni ayirish.

$$y=\sqrt{23}-3 y=-\sqrt{23}-3$$