Question

If -5 is a root of the quadratic equation 2x² + px - 15 = 0 and the quadratic equation P(x² + x) + k=0 has equal roots , find the value of k.

Answer 1

If -5 is a root of the quadratic equation 2x² + px - 15 = 0, hence we have

2(-5)² + p(-5) - 15 = 0

then p = 7

so the eq is 2x² + 7x - 15 = 0

its one root is -5 than other is (solving the eq) 3/2

P(x² + x) + k=0

=> 7(x² + x) + k=0
=>7x²+x+k=0
=>7(-5)²+(-5)+k=0
=>175-5+k=0
=>k=-170

Answer 2

Answer :

k = 7 / 4

Step-by-step explanation :

Given that ;

−5 is a root of the quadratic equation 2x²+px-15 = 0   …… (i)

And,

The quadratic equation p(x²+x)+k=0 has equal roots  … (ii)

As, the roots of given equation, the value of x = - 5, putting the values ;

2x² + px - 15 = 0

2 (-5)² + p (-5) - 15 = 0

⇒ 2 * 25 - 5p - 15 = 0

⇒ 50 - 5p - 15 = 0

⇒ 35 - 5p = 0

⇒ 5p = 35

⇒ p = 35 / 5

⇒ p = 7

Hence, the value of p is 7.

Now, on putting the value of p in eqⁿ (ii),

p ( x² + x ) + k = 0

7 ( x² + x ) + k = 0

7x² + 7x + k = 0

On comparing it with ax² + bx + c = 0,

a = 7

b = 7

c = k

D ( Discriminant ) = b² - 4ac

D = 7² - 4 * 7 * k

D = 49 - 28 k

Given that the quadratic equation has equal roots, i.e., D = 0

49 - 28k = 0

28k = 49

k = 49 / 28

∴ k = 7 / 4

Hence, the value of k is 7 / 4.