Question

Solve the following quadratic equations by factorization: $\frac{1}{(x-1)(x-2)} +\frac{1}{(x-2)(x-3)}+\frac{1}{(x-3)(x-4)}= \frac{1}{6}$

Answer 1

SOLUTION :  

Given : 1/(x−1)(x−2) +1/(x−2)(x−3) +1/(x−3)(x−4) = 1/6

[(x - 3) (x - 4) + (x - 1) (x - 4) + (x - 1) ( x - 2) ] / [(x - 1) (x - 2) (x - 3) (x - 4) ] = 1/6

[ By taking LCM]

[x² - 4x - 3x +12 + x² - 4x - 1x + 4 + x² - 2x - 1x +2] / [(x² - 5x + 4) (x² - 5x + 6] = ⅙

[3x² -7x - 5x -3x +18] /  [(x² - 5x + 4) (x² - 5x + 6] = ⅙

[3x² - 15x +18] /  [(x² - 5x + 4) (x² - 5x + 6] = ⅙

3[x² - 5x + 6] /  [(x² - 5x + 4) (x² - 5x + 6] = ⅙

3/[(x² - 5x + 4) = ⅙

3 × 6 = (x² - 5x + 4)

18 = (x² - 5x + 4)

x² - 5x + 4 - 18 = 0

x² - 5x - 14 = 0

x² - 7x + 2x - 14 = 0

x(x - 7) + 2 (x - 7 ) = 0

(x + 2) (x - 7) = 0

(x + 2)  = 0  or  (x - 7) = 0

x = - 2  or  x = 7  

Hence, the roots of the quadratic equation 1/(x−1)(x−2) +1/(x−2)(x−3) +1/(x−3)(x−4) = 1/6 are -2 & 7 .

★★ METHOD TO FIND SOLUTION OF a quadratic equation by FACTORIZATION METHOD :  

We first write the given quadratic polynomial as product of two linear factors by splitting the middle term and then equate each factor to zero to get desired roots of given quadratic equation.

HOPE THIS ANSWER WILL HELP YOU….

Answer 2

1/(x−1)(x−2) +1/(x−2)(x−3) +1/(x−3)(x−4) = 1/6

[(x - 3) (x - 4) + (x - 1) (x - 4) + (x - 1) ( x - 2) ] / [(x - 1) (x - 2) (x - 3) (x - 4) ] = 1/6

[ ʙʏ ᴛᴀᴋɪɴɢ ʟᴄᴍ]

[x² - 4x - 3x +12 + x² - 4x - 1x + 4 + x² - 2x - 1x +2] / [(x² - 5x + 4) (x² - 5x + 6] = ⅙

[3x² -7x - 5x -3x +18] /  [(x² - 5x + 4) (x² - 5x + 6] = ⅙

[3x² - 15x +18] /  [(x² - 5x + 4) (x² - 5x + 6] = ⅙

3[x² - 5x + 6] /  [(x² - 5x + 4) (x² - 5x + 6] = ⅙

3/[(x² - 5x + 4) = ⅙

3 × 6 = (x² - 5x + 4)

18 = (x² - 5x + 4)

x² - 5x + 4 - 18 = 0

x² - 5x - 14 = 0

x² - 7x + 2x - 14 = 0

x(x - 7) + 2 (x - 7 ) = 0

(x + 2) (x - 7) = 0

(x + 2)  = 0  ᴏʀ  (x - 7) = 0

x = - 2  ᴏʀ  x = 7  
ʜᴇɴᴄᴇ, ᴛʜᴇ ʀᴏᴏᴛs ᴏғ ᴛʜᴇ ǫᴜᴀᴅʀᴀᴛɪᴄ ᴇǫᴜᴀᴛɪᴏɴ 1/(x−1)(x−2) +1/(x−2)(x−3) +1/(x−3)(x−4) = 1/6 ᴀʀᴇ -2 & 7 .
★ ᴍᴇᴛʜᴏᴅ ᴛᴏ ғɪɴᴅ sᴏʟᴜᴛɪᴏɴ ᴏғ ᴀ ǫᴜᴀᴅʀᴀᴛɪᴄ ᴇǫᴜᴀᴛɪᴏɴ ʙʏ ғᴀᴄᴛᴏʀɪᴢᴀᴛɪᴏɴ ᴍᴇᴛʜᴏᴅ :  

ᴡᴇ ғɪʀsᴛ ᴡʀɪᴛᴇ ᴛʜᴇ ɢɪᴠᴇɴ ǫᴜᴀᴅʀᴀᴛɪᴄ ᴘᴏʟʏɴᴏᴍɪᴀʟ ᴀs ᴘʀᴏᴅᴜᴄᴛ ᴏғ ᴛᴡᴏ ʟɪɴᴇᴀʀ ғᴀᴄᴛᴏʀs ʙʏ sᴘʟɪᴛᴛɪɴɢ ᴛʜᴇ ᴍɪᴅᴅʟᴇ ᴛᴇʀᴍ ᴀɴᴅ ᴛʜᴇɴ ᴇǫᴜᴀᴛᴇ ᴇᴀᴄʜ ғᴀᴄᴛᴏʀ ᴛᴏ ᴢᴇʀᴏ ᴛᴏ ɢᴇᴛ ᴅᴇsɪʀᴇᴅ ʀᴏᴏᴛs ᴏғ ɢɪᴠᴇɴ ǫᴜᴀᴅʀᴀᴛɪᴄ ᴇǫᴜᴀᴛɪᴏɴ.