- Trending Categories
- 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

# Find the value of k such that the polynomial $x^{2}-(k+6)x+2(2k-1)$ has sum of its zeros equal to half of their product.

**Given:**Polynomial $x^{2}-(k+6)x+2(2k-1)$ has sum of its zeros equal to half of their product.

**To do:**To Find the value of $k$.

**Solution:**

Given Polynomial $x^{2}-(k+6)x+2(2k-1)$ is a quadratic polynomial.

On comparing it to 4ax^{2}+bx+c$ we have,

$a=1, b=-(k+6)$ and $2(2k-1)$

Let $\alpha$ and $\beta$ are the zeroes of the the given polynomial.

As known,

Sum of the zeroes, $\alpha+\beta=-b$

$\Rightarrow \alpha+\beta=-(-(k+6))=k+6$

Half of the Product of the zeroes, $\frac{\alpha.\beta}{2}=\frac{1}{2}×\frac{c}{a}=\frac{2(2k-1)}{2}=2k-1$

As given: $\alpha+\beta=\frac{\alpha.\beta}{2}$

$k+6=2k-1$

$\Rightarrow 2k-k=6+1$

$\Rightarrow k=7$

Therefore the value of $k=7$.

- Related Articles
- Polynomial \( f(x)=x^{2}-5 x+k \) has zeroes \( \alpha \) and \( \beta \) such that \( \alpha-\beta=1 . \) Find the value of \( 4 k \).
- Are the following statements 'True' or 'False'? Justify your answers.The only value of \( k \) for which the quadratic polynomial \( k x^{2}+x+k \) has equal zeros is \( \frac{1}{2} \)
- If the sum of the zeroes of the polynomial $P(x)=( k^{2}-14)x^{2}-2x-12$ is $1$. Then find the value of $k$.
- Find the value of $k$ for which the equation $x^{2}+k( 2x+k-1)+2=0$ has real and equal roots.
- If the zeros of the polynomial $f(x)\ =\ x^3\ -\ 12x^2\ +\ 39x\ +\ k$ are in A.P., find the value of $k$.
- Find the value of $k$ for which the following system of equations having infinitely many solutions: $2x\ +\ 3y\ =\ 2$ $(k\ +\ 2)x\ +\ (2k\ +\ 1)y\ =\ 2(k\ -\ 1)$
- Find the values of k for which the roots are real and equal in each of the following equations: $(2k+1)x^2 + 2(k+3)x + (k + 5) = 0$
- If sum of the square of zeroes of the polynomial $f(x)=x^2−8x+k$ is $40$, find the value of $k$.
- Find the value of \( k \), if \( x-1 \) is a factor of \( p(x) \) in each of the following cases:(i) \( p(x)=x^{2}+x+k \)(ii) \( p(x)=2 x^{2}+k x+\sqrt{2} \)(iii) \( p(x)=k x^{2}-\sqrt{2} x+1 \)(iv) \( p(x)=k x^{2}-3 x+k \)
- Find the values of k for which the quadratic equation $(3k + 1)x^2 + 2(k + 1)x + 1 = 0$ has equal roots. Also, find the roots.
- Find the values of k for which the quadratic equation $(k + 4)x^{2} + ( k+1)x+1=0 $ has equal roots. Also find these roots.
- Find the values of k for which the following equations have real and equal roots: $k^2x^2 - 2(2k - 1)x + 4 = 0$
- For what value of k, $(4 - k)x^2 + (2k + 4)x + (8k + 1) = 0$, is a perfect square.
- Find the values of k for which the following equations have real and equal roots: $x^2 - 2(k + 1)x + k^2 = 0$
- Find the values of k for which the following equations have real and equal roots: $(k+1)x^2 - 2(k - 1)x + 1 = 0$

Advertisements