3x^2=(x+√3)^2

题目

3x^2=(x+√3)^2
x^2+(√2 -1)x-2(3-2√2)=0
x^2-m(3x-2m+n)=n^2

答案

(1) 3x^2 = ( x + √3)^2
3x^2 = x^2 + 2√3 x + 3
2x^2 - 2√3 x - 3 = 0
方程判别式 △ = (-2√3)^2 - 4 * 2 * (-3) = 12 + 24 = 36
所以,方程的解是：　　x1 = ( 2√3 - √36)/(2 * 2) = (2√3 - 6)/4 = √3/2 - 3/2
x2 = √3/2 + 3/2
(2) x^2 - (√2 - 1) x - 2(3 - 2√2) = 0
解 方程的判别式△ = (√2 - 1)^2 - 4 * 1 * [ - 2(3 - 2√2)]
= 3 - 2√2 + 8(3 - 2√2)
= 3 - 2√2 + 24 - 16√2
= 27 - 18√2
所以：√△ = √（27 - 18√2） = √（18 - 18√2 + 9)
= √( 3√2 - 3)^2
= 3√2 - 3
所以,原方程的解是：x1 = ( 1 - √2 + 3 - 3√2)/2 = (4 - 4√2)/2 = 2 - 2√2
x2 = ( 1 - √2 - 3 + 3√2)/2 = (-2 + 2√2)/2 = √2 - 1
(3) x^2 - m(3x - 2m +n) = n^2
x^2 - 3mx + 2m^2 - mn = n^2
x^2 - 3mx + 2m^2 - mn - n^2 = 0
关于x的一元二次方程的判别式 △ = (-3m)^2 - 4(2m^2 - mn - n^2)
= 9m^2 - 8m^2 + 4mn + 4n^2
= m^2 + 4mn + 4n^2
所以 √△ = √(m^2 + 4mn + 4n^2) = √(m + 2n)^2 = m + 2n
因为,原方程的解是：x1 = ( -3m - m - 2n)/2 = -2m - n
x2 = ( -3m + m + 2n)/2 = -m + n