1=-1/2x^2+2x eching

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

x uchun yechish

$$x=\sqrt{2}+2\approx 3,414213562$$

$$x=2-\sqrt{2}\approx 0,585786438$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

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

Ikkala tarafdan 1 ni ayirish.

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

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

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

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\left(-\frac{1}{2}\right)\left(-1\right)}}{2\left(-\frac{1}{2}\right)}$$

-4 ni -\frac{1}{2} marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4+2\left(-1\right)}}{2\left(-\frac{1}{2}\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-2}}{2\left(-\frac{1}{2}\right)}$$

4 ni -2 ga qo'shish.

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

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

$$x=\frac{-2±\sqrt{2}}{-1}$$

x=\frac{-2±\sqrt{2}}{-1} tenglamasini yeching, bunda ± musbat. -2 ni \sqrt{2} ga qo'shish.

$$x=\frac{\sqrt{2}-2}{-1}$$

-2+\sqrt{2} ni -1 ga bo'lish.

$$x=2-\sqrt{2}$$

x=\frac{-2±\sqrt{2}}{-1} tenglamasini yeching, bunda ± manfiy. -2 dan \sqrt{2} ni ayirish.

$$x=\frac{-\sqrt{2}-2}{-1}$$

-2-\sqrt{2} ni -1 ga bo'lish.

$$x=\sqrt{2}+2$$

Tenglama yechildi.

$$x=2-\sqrt{2} x=\sqrt{2}+2$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

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

Ikkala tarafini -2 ga ko‘paytiring.

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

-\frac{1}{2} ga bo'lish -\frac{1}{2} ga ko'paytirishni bekor qiladi.

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

2 ni -\frac{1}{2} ga bo'lish 2 ga k'paytirish -\frac{1}{2} ga qaytarish.

$$x^{2}-4x=\frac{1}{-\frac{1}{2}}$$

1 ni -\frac{1}{2} ga bo'lish 1 ga k'paytirish -\frac{1}{2} ga qaytarish.

$$x^{2}-4x=-2$$

-4 ni bo‘lish, x shartining koeffitsienti, 2 ga -2 olish uchun. Keyin, -2 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-4x+\left(-2\right)^{2}=-2+\left(-2\right)^{2}$$

-2 kvadratini chiqarish.

$$x^{2}-4x+4=-2+4$$

-2 ni 4 ga qo'shish.

$$x^{2}-4x+4=2$$

x^{2}-4x+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.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x-2=\sqrt{2} x-2=-\sqrt{2}$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\sqrt{2}+2 x=2-\sqrt{2}$$