1. Find value of k for which `x^2-kx-4=0` has real roots
Solution:
`x^2-kx-4=0`
Comparing the given equation with `ax^2+bx+c=0`
We get `a=1,b=-k,c=-4`
The equation has real roots.
So, `Delta>=0`
`=>b^2-4ac>=0`
`=>(-k)^2-4*(1)*(-4)>=0`
`=>k^2+16>=0`
`=>k^2>=-16`
2. Find value of k for which quadratic equation `2x^2+kx+2=0` has equal roots
Solution:
`2x^2+kx+2=0`
Comparing the given equation with `ax^2+bx+c=0`
We get `a=2,b=k,c=2`
The equation has eq roots.
So, `Delta=0`
`=>b^2-4ac=0`
`=>(k)^2-4*(2)*(2)=0`
`=>k^2-16=0`
`=>k^2=16`
`=>k=+- 4`
The solution is
`k = 4,k = -4`
3. Find value of k for which quadratic equation `3x^2+11x+k=0` has reciprocal roots
Solution:
`3x^2+11x+k=0`
Comparing the given equation with `ax^2+bx+c=0`
We get `a=3,b=11,c=k`
The equation has reci roots.
So if one root is `alpha` then other root is `beta=1/(alpha)`
Product of roots `=alpha*beta=c/a`
`=>alpha*1/(alpha)=c/a`
`=>c/a=1`
`=>(k)/(3)=1`
`=>k=3`
4. Find value of k for which quadratic equation `x^2+kx+2=0` has sum of roots -3
Solution:
`x^2+kx+2=0`
Comparing the given equation with `ax^2+bx+c=0`
We get `a=1,b=k,c=2`
Sum of roots `=-3`
and Sum of roots `=alpha+beta=(-b)/a`
`:.(-b)/a=-3`
`=>(-k)/(1)=-3`
`=>-k=-3`
`=>k=3`
5. Find value of k for which quadratic equation `x^2+3x+k=0` has product of roots 2
Solution:
`x^2+3x+k=0`
Comparing the given equation with `ax^2+bx+c=0`
We get `a=1,b=3,c=k`
Product of roots `=2`
Product of roots `=alpha*beta=c/a`
`:.c/a=2`
`=>(k)/(1)=2`
`=>k=2`
6. Find value of k for which quadratic equation `x^2-(k+6)x+2(2k-1)=0` has sum of roots = `1/2` product of roots
Solution:
`x^2-(k+6)x+2(2k-1)=0`
`x^2-(k+6)x+4k-2=0`
Comparing the given equation with `ax^2+bx+c=0`
We get `a=1,b=-(k+6),c=4k-2`
Sum of roots `= 1/2` Product of roots
Sum of roots `=alpha+beta=(-b)/a`
`=(k+6)/1`
`=k+6`
Product of roots `=alpha*beta=c/a`
`=(4k-2)/(1)`
`=4k-2`
Sum of roots `= 1/2` Product of roots
`=>k+6=1/2*(4k-2)`
`=>k+6=2k-1`
`=>(k+6)-(2k-1)=0`
`=>k+6-2k+1=0`
`=>-k+7=0`
`=>-k=-7`
`=>k=7`
This material is intended as a summary. Use your textbook for detail explanation.
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