Question

If α and β are the zeros of the polynomial f (x) = x² - 5x + k such that α - β = 1, find the value of k.
a) 5
b) 6
c) 7
d) 8

Answer 1

✪ᴀɴsᴡᴇʀ✪

• $\star \star \large{\boxed{ (b) k = 6 }} \star \star$

★ɢɪᴠᴇɴ★

• $\alpha, \beta$ are zeroes of $f(x) = x^2 - 5x + k$
• $\alpha - \beta = 1$

✫ᴛᴏ ғɪɴᴅ✫

• Value of k.

᯽ғᴏʀᴍᴜʟᴀ᯽

• $|\alpha - \beta | = \frac{\sqrt{D}}{a}$ where D is the discriminant.

• $(\alpha + \beta )^2 - (\alpha - \beta )^2 = 4\alpha \beta$

⍟ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴ⍟

Method I

Here,

$\implies \alpha + \beta = 5,\:\alpha \beta = k$

Now,

$\implies (\alpha + \beta )^2 - (\alpha - \beta )^2 = 4\alpha \beta$

$\implies 5^2 - 1^2 = 4k$

$\implies 25 - 1 = 4k$

$\implies k = \frac{24}{4}$

$\implies k = 6$

Method II

Here,

$\implies |\alpha - \beta | = \frac{\sqrt{b^2 - 4ac}} {a}$

$\implies 1 = \frac{\sqrt{(-5)^2 - 4.1.k}}{1}$

$\implies 1 = 25 - 4k$

$\implies 4k = 25 - 1$

$\implies 4k = 24$

$\implies k = \frac{24}{4}$

$\implies k = 6$