- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- MS Excel
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP
- Physics
- Chemistry
- Biology
- Mathematics
- English
- Economics
- Psychology
- Social Studies
- Fashion Studies
- Legal Studies
- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who
Write all the values of k for which the quadratic equation $x^2+kx+16=0$ has equal roots. Find the roots of the equation so obtained.
Given:
Given quadratic equation is $x^2 + kx + 16 = 0$.
To do:
We have to find the value of k for which the given quadratic equation has equal roots.
Solution:
$x^2 + kx + 16 = 0$
Comparing the given quadratic equation with the standard form of the quadratic equation $ax^2+bx+c=0$, we get,
$a=1, b=k$ and $c=16$.
The discriminant of the standard form of the quadratic equation $ax^2+bx+c=0$ is $D=b^2-4ac$.
$D=(k)^2-4(1)(16)$
$D=k^2-64$
The given quadratic equation has equal roots if $D=0$.
Therefore,
$k^2-64=0$
$k^2-(8)^2=0$
$(k+8)(k-8)=0$
$k+8=0$ or $k-8=0$
$k=-8$ or $k=8$
The values of k are $-8$ and $8$.
For $k = -8$,
$x^2 + kx + 16 = 0$
$x^2 + (-8)x + 16 = 0$
$x^2 - 8x + 16 = 0$
$(x - 4)^2 = 0$
$x-4=0$
$x=4$
Therefore, for $k=-8$ the roots of the given quadratic equation are $4$ and $4$.
For $k = 8$,
$x^2 + kx + 16 = 0$
$x^2 + 8x + 16 = 0$
$(x + 4)^2 = 0$
$x+4=0$
$x=-4$
Therefore, for $k=8$ the roots of the given quadratic equation are $-4$ and $-4$.