Question

For what value of k the zeros of x²-5x+k differ by 1 ?
Please give a step by step explanation!!

Answer 1

Gɪᴠᴇɴ :-

• The Difference b/w zeros of x² - 5x + k = 1

Tᴏ Fɪɴᴅ :-

• value of k ?

ᴄᴏɴᴄᴇᴘᴛ ᴜsᴇᴅ :-

→ The sum of the roots of the Equation ax² + bx + c = 0 , is given by = (-b/a)

and ,

→ Product of roots of the Equation is given by = c/a.

Sᴏʟᴜᴛɪᴏɴ :-

comparing the given Polynomial x² - 5x + k = 0 with ax² + bx + c = 0 , we get,

→ a = 1

→ b = (-5)

→ c = k

Let us Assume That, Two Zeros of the Given Polynomial are ɑ & β .

Than,

➻ ɑ + β = (-b/a) = -(-5)/1

➻ ɑ + β = 5

➻ ɑ - β = 1 (Given).

Adding Both We get,

➻ (ɑ + β) + (ɑ - β) = 5 + 1

➻ 2ɑ = 6

➻ ɑ = 3

So,

➻ β = 5 - 3 = 2 .

Therefore,

➻ ɑ * β = (c/a)

➻ ɑ * β = k/1

➻ ɑ * β = k

➻ 3 * 2 = k

➻ k = 6 (Ans.)

Hence, Value of k will be 6.

Answer 2

Answer:-

Given Polynomial => x² - 5x + k = 0

Let a = 1 ; b = - 5 ; c = k and the roots be p , q.

We know that,

Sum of the roots = - b/a

→ p + q = - (- 5)/1

→ p + q = 5 -- equation (1).

Product of the roots = c/a

→ pq = k/1

→ pq = k -- equation (2).

Difference of the roots = 1

→ p - q = 1 -- equation (3)

We know that,

(a - b)² = (a + b)² - 4ab

→ (p - q)² = (p + q)² - 4pq

→ (1)² = (5)² - 4(k)

→ 1 = 25 - 4k

→ 4k = 25 - 1

→ 4k = 24

→ k = 24/4

→ k = 6

Hence, the value of k is 6.