Maths

QUADRATIC EQUATIONS 79
So, 9x 2 – 15x + 6 = 0 can be written as
2
5✁
1
3x
✂ ✂ ☎
✝ 2 4
=0
✄
✆
i.e.,
5 ✞ ✟
✠ ✡ ☛ 3x
2
2
= 1 4
☞
So, the solutions of 9x 2 – 15x + 6 = 0 are the same as those of
2
5✁
1
3x
✂ ✍ ☎ .
2 4
✌
✄
✆
✝
i.e., 3x – 5 2 = 1 2 or 5
3x ✎ =
2
1
✎
2
(We can also write this as
5 1
3x ✏ ✑ ✒ , where ‘✓’ denotes ‘plus minus’.)
2 2
Thus, 3x = 5 1 ✔ or
2 2
So, x = 5 1 ✖ or
6 6
5 1
3x ✕ ✎
2 2
5 1
x ✑ ✏
6 6
Therefore, x = 1 or x = 4 6
i.e., x = 1 or x = 2 3
Therefore, the roots of the given equation are 1 and 2 .
3
Remark : Another way of showing this process is as follows :
The equation 3x 2 – 5x + 2 = 0
is the same as
2 5 2
x x ✏ ✖ =0
3 3
Now, x 2 – 5 x 2 ✖ =
3 3
2 2
1 5 1 5 2
✗ ✘
✘ ✙ ✗ ✚ ✙ ✚
✛ ✛ ✜
✢ ✣ ✢ ✣
✤ ✥ ✤ ✥
x
✧ ✦ ✧
★ ✩ ★ ✩
✦ 2 3 2 3 3