Question

if the difference between the roots of the equation x^2-13x+k=0 is 17 find k

Answer 1

Step-by-step explanation:

Given -

• If the difference between the roots of the equation x² - 13x + k = 0 is 17

To Find -

• Value of k

Now,

• α - β = 17 (given) ...... (i)

And

As we know that :-

→ α + β = -b/a

→ α + β = 13 .......(ii)

Now, By solving (i) and (ii), we get :

→ α + β = 13

α - β = 17

_________

→ 2α = 30

→ α = 30/2

→ α = 15

Now, Substituting the value of α on (ii), we get :

→ α + β = 13

→ β = 13 - 15

→ β = -2

Then,

The value of α is 15 and β is -2

Now,

The value of k is

• αβ = c/a = k

→ 15 × -2 = k

→ k = -30

Hence,

The value of k is -30

Verification :-

→ x² - 13x - 30

→ x² + 2x - 15x - 30

→ x(x + 2) - 15(x + 2)

→ (x - 15)(x + 2)

Zeroes are -

→ x - 15 = 0 and x + 2 = 0

→ x = 15 and x = -2

Then,

→ α - β = 17

→ 15 - (-2) = 17

→ 15 + 2 = 17

→ 17 = 17

LHS = RHS

Hence,

Verified...

It shows that our answer is absolutely correct.

Answer 2

Given ,

The quadratic eq is x² - 13x + k = 0

and

The difference between the roots of quadratic equation is 17 i.e

$\sf \Rightarrow α - β = 17 \: --- \: (i)$

We know that , the sum of roots i.e α + β is

$\sf \star \: \: \alpha + \beta = - \frac{b}{a}$

Thus ,

$\sf \Rightarrow α + β = 13 \: - - - \: (ii)$

Add eq (i) and (ii) , we get

$\sf \Rightarrow 2α = 30 \\ \\ \sf \Rightarrow α = \frac{30}{2} </p><p>\\ \\ \sf \Rightarrow α = 15$

Put the value of α = 15 in eq (ii) , we get

$\sf \Rightarrow 15 + β = 13 \\ \\ \sf \Rightarrow</p><p>β = 13 - 15 \\ \\ \sf \Rightarrow </p><p>β = -2$

Now , the product of roots i.e α × β is

$\sf \star \: \: \alpha \times \beta = \frac{c}{a}$

Thus ,

$\sf \Rightarrow 15 × -2 = k \\ \\ \sf \Rightarrow </p><p>k = -30</p><p>$

$\therefore \underline{ \sf \bold{The \: value \: of \: k \: is -30}}$