14x+10.5=x^2+1.5x eching

$$14 x + 10.5 = x ^ { 2 } + 1.5 x$$

x uchun yechish

$$x = \frac{\sqrt{793} + 25}{4} \approx 13,29006392$$

$$x=\frac{25-\sqrt{793}}{4}\approx -0,79006392$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan x^{2} ni ayirish.

$$14x+10,5-x^{2}=1,5x$$

Ikkala tarafdan 1,5x ni ayirish.

$$14x+10,5-x^{2}-1,5x=0$$

12,5x ni olish uchun 14x va -1,5x ni birlashtirish.

$$12,5x+10,5-x^{2}=0$$

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.

$$-x^{2}+12,5x+10,5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 12,5 ni b va 10,5 ni c bilan almashtiring.

$$x=\frac{-12,5±\sqrt{12,5^{2}-4\left(-1\right)\times 10,5}}{2\left(-1\right)}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib 12,5 kvadratini chiqarish.

$$x=\frac{-12,5±\sqrt{156,25-4\left(-1\right)\times 10,5}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-12,5±\sqrt{156,25+4\times 10,5}}{2\left(-1\right)}$$

4 ni 10,5 marotabaga ko'paytirish.

$$x=\frac{-12,5±\sqrt{156,25+42}}{2\left(-1\right)}$$

156,25 ni 42 ga qo'shish.

$$x=\frac{-12,5±\sqrt{198,25}}{2\left(-1\right)}$$

198,25 ning kvadrat ildizini chiqarish.

$$x=\frac{-12,5±\frac{\sqrt{793}}{2}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-12,5±\frac{\sqrt{793}}{2}}{-2}$$

x=\frac{-12,5±\frac{\sqrt{793}}{2}}{-2} tenglamasini yeching, bunda ± musbat. -12,5 ni \frac{\sqrt{793}}{2} ga qo'shish.

$$x=\frac{\sqrt{793}-25}{-2\times 2}$$

\frac{-25+\sqrt{793}}{2} ni -2 ga bo'lish.

$$x=\frac{25-\sqrt{793}}{4}$$

x=\frac{-12,5±\frac{\sqrt{793}}{2}}{-2} tenglamasini yeching, bunda ± manfiy. -12,5 dan \frac{\sqrt{793}}{2} ni ayirish.

$$x=\frac{-\sqrt{793}-25}{-2\times 2}$$

\frac{-25-\sqrt{793}}{2} ni -2 ga bo'lish.

$$x=\frac{\sqrt{793}+25}{4}$$

Tenglama yechildi.

$$x=\frac{25-\sqrt{793}}{4} x=\frac{\sqrt{793}+25}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan x^{2} ni ayirish.

$$14x+10.5-x^{2}=1.5x$$

Ikkala tarafdan 1.5x ni ayirish.

$$14x+10.5-x^{2}-1.5x=0$$

12.5x ni olish uchun 14x va -1.5x ni birlashtirish.

$$12.5x+10.5-x^{2}=0$$

Ikkala tarafdan 10.5 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$12.5x-x^{2}=-10.5$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-x^{2}+12.5x=-10.5$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+12.5x}{-1}=-\frac{10.5}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{12.5}{-1}x=-\frac{10.5}{-1}$$

12.5 ni -1 ga bo'lish.

$$x^{2}-12.5x=-\frac{10.5}{-1}$$

-10.5 ni -1 ga bo'lish.

$$x^{2}-12.5x=10.5$$

-12.5 ni bo‘lish, x shartining koeffitsienti, 2 ga -6.25 olish uchun. Keyin, -6.25 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-12.5x+\left(-6.25\right)^{2}=10.5+\left(-6.25\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -6.25 kvadratini chiqarish.

$$x^{2}-12.5x+39.0625=10.5+39.0625$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali 10.5 ni 39.0625 ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-12.5x+39.0625=49.5625$$

x^{2}-12.5x+39.0625 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-6.25\right)^{2}=49.5625$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-6.25\right)^{2}}=\sqrt{49.5625}$$

Qisqartirish.

$$x-6.25=\frac{\sqrt{793}}{4} x-6.25=-\frac{\sqrt{793}}{4}$$

6.25 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{793}+25}{4} x=\frac{25-\sqrt{793}}{4}$$