F.4 Maths 問題,Quadratic Equations In One Unknown

只是給予你答案只是害了你
Quadratic Equations In One Unknown最重要的是了解delta的性質
相同根(只有一個根) 則 delta = 0
有兩個不等根 則delta > 0
沒有實根 則delta < 0
以上數題都是考驗你對delta的性質的了解


1.

x^2 + 4kx + 16k + 20 = 0

題目字眼 : a doubel real root 則delta = 0
delta = b²-4ac=0
(4k)² - 4(16k+20) = 0
16k²-64k-80 = 0
k=5 or k=-1

2a


x^2 - 2kx + 2k + 15= 0 has equal real roots
題目字眼 : has equal real roots 則delta = 0
delta = b²-4ac=0
(-2k)²- 4(2k+15) = 0
4k²-8k -60 = 0
k=5 or k=-3

2b

若然k=5,
x² - 2(5)x + 2(5) +15 = 0
x² -10x+25=0
當我們使用交叉相乘來因式分解的時候
出到的答案是一個完美平方
即(x-5)² = (x-5)(x-5)
(x-5)(x-5) = 0
x-5=0
x=5 (只有此實根)

若然 k=-3
x² - 2(-3)x + 2(-3) +15 = 0

當我們使用交叉相乘來因式分解的時候
出到的答案是一個完美平方

即(x+3)² = (x+3)(x+3)
(x+3)(x+3)=0
x+3=0
x=-3(repeated)

1.

x^2 + 4kx + 16k + 20 = 0

it has a repeated real root only
then delta=0
(4k)² - 4(16k+20) = 0
16k²-64k-80 = 0
k=5 or k=-1

2a


x^2 - 2kx + 2k + 15= 0 has equal real roots
then delta=0
(-2k)²- 4(2k+15) = 0
4k²-8k -60 = 0
k=5 or k=-3

2b


for k=5,
x² - 2(5)x + 2(5) +15 = 0
x=5 (repeated)
for k=-3
x² - 2(-3)x + 2(-3) +15 = 0
x=-3(repeated)