Question

If (-5) is a root of the quadratic equation 2x² + pa: + 15 = 0 and quadratic equation p(x² + x)+k= 0 has equal roots, then find the values of p and k​

Answer 1

Step-by-step explanation:

Correction :-

If (-5) is a root of the quadratic equation

2x² + px + 15 = 0 and quadratic equation

p(x² + x)+k= 0 has equal roots, then find the values of p and k ?

Solution:-

Given quadratic equation is

2x²+ px + 15 = 0

Given root = -5

If -5 is the root of the given equation then it satisfies the given equation.

=> 2(-5)²+p(-5)+15 = 0

=> 2(25)+(-5p)+15 = 0

=> 50-5p +15 = 0

=> 65-5p = 0

=> 5p = 65

=> p = 65/5

=> p = 13

Therefore,the value of p = 13

If p = 13 then the given equation becomes 2x²+13x+15 = 0

Given equation = p(x² + x)+k= 0

On Substituting the value of p in the equation then it will be 13(x²+x)+k = 0

=> 13x²+13x+k = 0

On Comparing this with the standard quadratic equation ax²+bx+c = 0

a = 13

b = 13

c = k

Given that

It has equal roots then its discriminant must be equal to zero.

The discriminant (D)= b²-4ac

We have ,

D = b²-4ac = 0

On Substituting these values in the above formula then

=> (13)²-4(13)(k) = 0

=> 169-52k = 0

=> 169 = 52k

=> 52k = 169

=> k = 169/52

=>k = 13/4

Therefore,k = 13/4

Answer:-

The value of p for the given problem is 13

The value of k for the given problem is 13/4

Used formulae:-

• The standard quadratic equation is ax²+bx+c = 0
• The discriminant (D)= b²-4ac
• An equation has equal roots then its discriminant must be equal to zero.

Points to know:-

The discriminant of ax²+bx+c = 0 is D= b²-4ac

• If D>0 it has two distinct and real roots.
• If D< 0 it has no real roots i.e.imaginary.
• If D = 0 it has real and equal roots.