Question

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1.) Find the value of p, for which one of the roots of the equation $\sf px²-14x+8=0$ is 6 times the other

2.) if α & β are the roots of $\sf x²+5x+a = 0$ and $\sf 2α+5β = -1$ find the value of a.

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Answer both of them ✔️

Step by step explanation ✅

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Answer 1

Answer:

1) p = 3

2) a = –24

Step-by-step explanation:

TO KNOW :

For a quadratic equation of the form ax² + bx + c = 0 ;

sum of roots = –(x coefficient)/x² coefficient = –b/a

product of roots = constant/x² coefficient = c/a

1) Given :

One of the roots of the equation px²-14x+8 = 0 is 6 times the other

To find :

the value of p

Solution :

Let one root be a

then the other root = 6a

For the given quadratic equation,

x² coefficient = p

x coefficient = –14

constant = 8

Sum of roots :

a + 6a = –(–14)/p

7a = 14/p

a = 14/7p

a = 2/p

Product of roots :

a × 6a = 8/p

6a² = 8/p

6(2/p)² = 8/p

6 × (4/p²) = 8/p

24/p² = 8/p

24/p = 8

p = 24/8

p = 3

The value of p is 3

_______________________

2) Given :

• α & β are the roots of x²+5x+a = 0 and
• 2α+5β = -1

To find :

the value of a

Solution :

For the given quadratic equation,

x² coefficient = 1

x coefficient = 5

constant = a

Sum of roots :

α + β = –5/1

α + β = –5

Product of roots :

αβ = a

2α + 5β = –1

2α + 2β + 3β = –1

2(α + β) + 3β = –1

2(–5) + 3β = –1

–10 + 3β = –1

3β = –1 + 10

3β = 9

β = 9/3

β = 3

α + β = –5

α + 3 = –5

α = –5 – 3

α = –8

αβ = a

(–8)(3) = a

a = –24

The value of a is –24

Answer 2

1. Hence the Answer is p = 3.

2. Answer value of a = -24.

Step-by-step explanation:

Note : Explanation is written in the attachment.