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
SOLUTION :
Given : (- 5) is the root of quadratic equation 2x² + px – 15 = 0 ………….(1)
& p(x² + x ) + k = 0 has Equal roots………….(2)
On putting the value of given root i.e x = - 5 in eq 1 .
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 = 7
p = 7
Hence the value of p is 7.
On putting the value of p = 7 in eq 2,
p(x² + x ) + k = 0
7(x² + x ) + k = 0
7x² + 7x + k = 0
On comparing the given equation with ax² + bx + c = 0
Here, a = 7, b = 7 and c = k
D(discriminant) = b² – 4ac
Given : Quadratic equation has equal roots i.e D = 0
b² – 4ac = 0
7² – 4(7)(k) = 0
49 – 28k = 0
49 = 28k
k = 49/ 28 = 7/4
k = 7 /4
Hence, the value of k = is 7/4 .
★★ NATURE OF THE ROOTS
If D = 0 roots are real and equal
If D > 0 roots are real and distinct
If D < 0 No real roots
HOPE THIS ANSWER WILL HELP YOU…
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