中四數學題

1.已知二次函數f(x) = x^2-4x+1,且f(a) = f(b) = 0,其中a≠b
a.計算a^2-4a+1及b^2-4b+1 b.由此,求a+b的值 c.求f(a+b)
(a)
a^2-4a+1=f(a)=0
b^2-4b+1=f(b)=0
(b)
a和b是f(x)的根
a+b=4
(c)
f(a+b)=f(4)=1
2.a.展開(a+b)(a-b) b.計算(√21-√5)(√21+√5)的值
c.log２ (√21-√5)+log２ (√21+√5)的值
(a)
(a+b)(a-b)
=a^2+ab-ab-b^2
=a^2-b^2
(b)
代a=√21,b=√5
(√21-√5)(√21+√5)
=(√21)^2-(√5)^2
=21-5
=16
(c)
log２ (√21-√5)+log２ (√21+√5)
=log2[(√21-√5)+(√21+√5)]
=2√21log2
3. 已知方程2^x+1=5^x-1...............(* )
a.把方程(*)改寫a^x=10的形式 b.由此,解方程2^x+1=5^x-1(答案須準確至三位有效數字)
(a)
2^(x+1)=5^(x-1)
2(2^x)=(5^x)/5
10=(5/2)^x
log10=xlog(5/2)
1=xlog(5/2)
x=2.513 (答案準確至三位有效數字)
4.a因式分解a^x+1 - a^x b.由此,解方程6^x+1 - 6^x = 5/6
(a)
a^(x+1) - a^x
=a^x(a-1)
(b)
6^(x+1) - 6^x = 5/6
6^x(6-1) = 5/6
6^x(5) = 5/6
6^(x+1)=5
(x+1)log6=log5
x=-0.102
5.已知y= 3x^2+ax - 5
a.若3x^2+ax-5 ≡ 3(x-1)^2+b,求a及b的值
b由此,求y的最小值
(a)
3x^2+ax-5 ≡ 3(x-1)^2+b
3x^2+ax-5 ≡ 3(x^2-2x+1)+b
3x^2+ax-5 ≡ 3x^2-6x+3+b
a=-6
-5=3+b
b=-8
(b)
y
= 3x^2+ax - 5
= 3x^2-6x - 5
=3(x-1)^2-8
y的最小值是-8