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-x^{2}+90x-75=20
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}+90x-75-20=20-20
Tenglamaning ikkala tarafidan 20 ni ayirish.
-x^{2}+90x-75-20=0
O‘zidan 20 ayirilsa 0 qoladi.
-x^{2}+90x-95=0
-75 dan 20 ni ayirish.
x=\frac{-90±\sqrt{90^{2}-4\left(-1\right)\left(-95\right)}}{2\left(-1\right)}
Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 90 ni b va -95 ni c bilan almashtiring.
x=\frac{-90±\sqrt{8100-4\left(-1\right)\left(-95\right)}}{2\left(-1\right)}
90 kvadratini chiqarish.
x=\frac{-90±\sqrt{8100+4\left(-95\right)}}{2\left(-1\right)}
-4 ni -1 marotabaga ko'paytirish.
x=\frac{-90±\sqrt{8100-380}}{2\left(-1\right)}
4 ni -95 marotabaga ko'paytirish.
x=\frac{-90±\sqrt{7720}}{2\left(-1\right)}
8100 ni -380 ga qo'shish.
x=\frac{-90±2\sqrt{1930}}{2\left(-1\right)}
7720 ning kvadrat ildizini chiqarish.
x=\frac{-90±2\sqrt{1930}}{-2}
2 ni -1 marotabaga ko'paytirish.
x=\frac{2\sqrt{1930}-90}{-2}
x=\frac{-90±2\sqrt{1930}}{-2} tenglamasini yeching, bunda ± musbat. -90 ni 2\sqrt{1930} ga qo'shish.
x=45-\sqrt{1930}
-90+2\sqrt{1930} ni -2 ga bo'lish.
x=\frac{-2\sqrt{1930}-90}{-2}
x=\frac{-90±2\sqrt{1930}}{-2} tenglamasini yeching, bunda ± manfiy. -90 dan 2\sqrt{1930} ni ayirish.
x=\sqrt{1930}+45
-90-2\sqrt{1930} ni -2 ga bo'lish.
x=45-\sqrt{1930} x=\sqrt{1930}+45
Tenglama yechildi.
-x^{2}+90x-75=20
Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.
-x^{2}+90x-75-\left(-75\right)=20-\left(-75\right)
75 ni tenglamaning ikkala tarafiga qo'shish.
-x^{2}+90x=20-\left(-75\right)
O‘zidan -75 ayirilsa 0 qoladi.
-x^{2}+90x=95
20 dan -75 ni ayirish.
\frac{-x^{2}+90x}{-1}=\frac{95}{-1}
Ikki tarafini -1 ga bo‘ling.
x^{2}+\frac{90}{-1}x=\frac{95}{-1}
-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.
x^{2}-90x=\frac{95}{-1}
90 ni -1 ga bo'lish.
x^{2}-90x=-95
95 ni -1 ga bo'lish.
x^{2}-90x+\left(-45\right)^{2}=-95+\left(-45\right)^{2}
-90 ni bo‘lish, x shartining koeffitsienti, 2 ga -45 olish uchun. Keyin, -45 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.
x^{2}-90x+2025=-95+2025
-45 kvadratini chiqarish.
x^{2}-90x+2025=1930
-95 ni 2025 ga qo'shish.
\left(x-45\right)^{2}=1930
x^{2}-90x+2025 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.
\sqrt{\left(x-45\right)^{2}}=\sqrt{1930}
Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.
x-45=\sqrt{1930} x-45=-\sqrt{1930}
Qisqartirish.
x=\sqrt{1930}+45 x=45-\sqrt{1930}
45 ni tenglamaning ikkala tarafiga qo'shish.