Question

If α, β are the roots of x2 + px + 23 = 0 and α − β = 1 then find p​

Answer 1

Answer :

p = √93

Note:

★ The possible values of the variable which satisfy the equation are called its roots or solutions .

★ A quadratic equation can have atmost two roots .

★ The general form of a quadratic equation is given as ; ax² + bx + c = 0

★ If α and ß are the roots of the quadratic equation ax² + bx + c = 0 , then ;

• Sum of roots , (α + ß) = -b/a

• Product of roots , (αß) = c/a

Solution :

Here ,

The given quadratic equation is ;

x² + px + 23 = 0 .

Now ,

Comparing the given quadratic equation with the general quadratic equation ax² + bx + c = 0 , we have ;

a = 1

b = p

c = 23

Also ,

It is given that , α and ß are the roots of the given quadratic equation .

Thus ,

=> Sum of roots = -b/a

=> α + ß = -p/1

=> α + ß = -p

Also ,

=> Product of roots = c/a

=> αß = 23/1

=> αß = 23

Also ,

It is given that ,

α - ß = 1

Now ,

We know that , (a + b)² = (a - b)² + 4ab

Thus ,

=> (α + ß)² = (α - ß)² + 4αß

=> (-p)² = 1² + 4•23

=> p² = 1 + 92

=> p² = 93

=> p = √93

Hence , p = √93 .

Answer 2

Answer:

p=√93

Step-by-step explanation:

x2+px+23 = 0

ø-ß=1

p=?

ø+ß = √(ø-ß)+4øß

ø+ß = √(1)+(c/a)

ø+ß = √1+4(23)

ø+ß = ±√93

ø+ß = -b/a

±√93 = -p/1

p = ±√93