2x(x-5)+3x=10(1/2-x) eching

$$2 x ( x - 5 ) + 3 x = 10 ( \frac { 1 } { 2 } - x )$$

x uchun yechish

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2,5$$

$$x=1$$

Kvadrat tenglamalaridan foydalanish qoidalari

2x ga x-5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-10x+3x=10\left(\frac{1}{2}-x\right)$$

-7x ni olish uchun -10x va 3x ni birlashtirish.

$$2x^{2}-7x=10\left(\frac{1}{2}-x\right)$$

10 ga \frac{1}{2}-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-7x=10\times \frac{1}{2}-10x$$

\frac{10}{2} hosil qilish uchun 10 va \frac{1}{2} ni ko'paytirish.

$$2x^{2}-7x=\frac{10}{2}-10x$$

5 ni olish uchun 10 ni 2 ga bo‘ling.

$$2x^{2}-7x=5-10x$$

Ikkala tarafdan 5 ni ayirish.

$$2x^{2}-7x-5=-10x$$

10x ni ikki tarafga qo’shing.

$$2x^{2}-7x-5+10x=0$$

3x ni olish uchun -7x va 10x ni birlashtirish.

$$2x^{2}+3x-5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, 3 ni b va -5 ni c bilan almashtiring.

$$x=\frac{-3±\sqrt{3^{2}-4\times 2\left(-5\right)}}{2\times 2}$$

3 kvadratini chiqarish.

$$x=\frac{-3±\sqrt{9-4\times 2\left(-5\right)}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9-8\left(-5\right)}}{2\times 2}$$

-8 ni -5 marotabaga ko'paytirish.

$$x=\frac{-3±\sqrt{9+40}}{2\times 2}$$

9 ni 40 ga qo'shish.

$$x=\frac{-3±\sqrt{49}}{2\times 2}$$

49 ning kvadrat ildizini chiqarish.

$$x=\frac{-3±7}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-3±7}{4}$$

x=\frac{-3±7}{4} tenglamasini yeching, bunda ± musbat. -3 ni 7 ga qo'shish.

$$x=\frac{4}{4}$$

4 ni 4 ga bo'lish.

$$x=1$$

x=\frac{-3±7}{4} tenglamasini yeching, bunda ± manfiy. -3 dan 7 ni ayirish.

$$x=-\frac{10}{4}$$

\frac{-10}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{5}{2}$$

Tenglama yechildi.

$$x=1 x=-\frac{5}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

2x ga x-5 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-10x+3x=10\left(\frac{1}{2}-x\right)$$

-7x ni olish uchun -10x va 3x ni birlashtirish.

$$2x^{2}-7x=10\left(\frac{1}{2}-x\right)$$

10 ga \frac{1}{2}-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2x^{2}-7x=10\times \frac{1}{2}-10x$$

\frac{10}{2} hosil qilish uchun 10 va \frac{1}{2} ni ko'paytirish.

$$2x^{2}-7x=\frac{10}{2}-10x$$

5 ni olish uchun 10 ni 2 ga bo‘ling.

$$2x^{2}-7x=5-10x$$

10x ni ikki tarafga qo’shing.

$$2x^{2}-7x+10x=5$$

3x ni olish uchun -7x va 10x ni birlashtirish.

$$2x^{2}+3x=5$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}+3x}{2}=\frac{5}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{3}{2}x=\frac{5}{2}$$

\frac{3}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3}{4} olish uchun. Keyin, \frac{3}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=\frac{5}{2}+\left(\frac{3}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{4} kvadratini chiqarish.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{5}{2}+\frac{9}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{2} ni \frac{9}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{49}{16}$$

x^{2}+\frac{3}{2}x+\frac{9}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{3}{4}\right)^{2}=\frac{49}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Qisqartirish.

$$x+\frac{3}{4}=\frac{7}{4} x+\frac{3}{4}=-\frac{7}{4}$$

Tenglamaning ikkala tarafidan \frac{3}{4} ni ayirish.

$$x=1 x=-\frac{5}{2}$$