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.
Answers
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.