Px^2-14x+8=0
let  a and b be the roots of the equation
b=6a
sum of roots = -b/a
product of roots = c/a
here a=p , b=-14 , c=8
a+b=14/p
ab=8/p


a+6a=14/p
7a=14/p
a=2/p

a*6a=8/p
6a^2=8/p
3a^2=4/p


3*(2/p)^2=4/p

3*4/p^2=4/p

p=3


I hope it's helpful for us.....

Introduction:

In this problem, we are given a quadratic equation in the form of px^2 - 14x + 8 = 0 and we need to find the value of p for which one root of the equation is 6 times the other.

Method:

To solve the problem, we will use the quadratic formula to find the roots of the equation. Once we have the roots, we can use the given information to set up an equation in terms of p and solve for it.

Step 1: Find the Roots of the Quadratic Equation:

Using the quadratic formula, we can find the roots of the equation px^2 - 14x + 8 = 0 as follows:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

Here, a = p, b = -14, and c = 8. Plugging in these values, we get:

x = (14 ± sqrt(196 - 32p)) / 2p

Step 2: Use the Given Information to Set Up an Equation:

We are given that one root is 6 times the other. Let's call the larger root y and the smaller root z. Then we have:

y = 6z

We also know that the sum of the roots is given by:

y + z = 14/p

Substituting y = 6z into the above equation, we get:

7z = 14/p

Solving for z, we get:

z = 2/p

Substituting z = 2/p into y = 6z, we get:

y = 12/p

Step 3: Solve for p:

Now we have expressions for both roots in terms of p. Substituting these into the quadratic formula, we get:

(14 ± sqrt(196 - 32p)) / 2p = 2/p or (14 ± sqrt(196 - 32p)) / 2p = 12/p

Simplifying, we get:

14p ± sqrt(196 - 32p) = 4 or 14p ± sqrt(196 - 32p) = 6

Squaring both sides and simplifying, we get:

(196 - 32p) = (4 - 14p)^2 or (196 - 32p) = (6 - 14p)^2

Expanding and simplifying, we get:

p^2 - 2p - 15 = 0 or p^2 - 4p + 5 = 0

Solving each equation, we get:

p = -3 or p = 1 or p = 5

Conclusion:

Therefore, the value of p for which one root of the quadratic equation px^2 - 14x + 8 = 0 is 6 times the other is p = 5.