Question

if the ratio of the roots of x^2+kx+27=0 is 1 : 3, then k =

Answer 1

ANSWER IS 12 AND -12

See explanation from the image

Answer 2

Step-by-step explanation:

Given:-

The ratio of the roots of x^2+kx+27=0 is 1 : 3

To find:-

Find the value of k ?

Solution:-

Given quadratic equation = x^2+kx+27=0

On comparing with the standard quadratic equation ax^2+bx+c = 0

a = 1

b = k

c = 27

The ratio of the roots = 1:3

Let they be X and 3X

we know that

Sum of the roots (α+ β) = -b/a

=> X+3X = -k/1

=> 4X = -k

=> k = -4X -------(1)

Product of the roots (αβ)=c/a

=>X(3X) = 27/1

=> 3X^2 = 27

=>X^2 = 27/3

=>X^2 = 9

=>X = ±√9

=> X = ±3

If X = 3 then k = -4(3) = -12

If X =-3 then k = -4(-3) = 12

Answer:-

The value of k for the given problem is 12 or -12

Check:-

I) If k = 12 then the equation is x^2+12x+27=0

=>x^2+3x+9x+27=0

=>x(x+3)+9(x+3) = 0

=>(x+3)(x+9)=0

=>x+3 = 0 or x+9 = 0

=> x= -3 or x= -9

The roots are -3 and -9

Their ratio = -3:-9 = 3:9 = 1:3

ii) If k = -12 then the equation is x^2-12x+27=0

=>x^2-3x-9x+27=0

=>x(x-3)-9(x+3) = 0

=>(x-3)(x-9)=0

=>x-3 = 0 or x+9 = 0

=> x= 3 or x= 9

The roots are 3 and 9

Their ratio = 3:9 = 1:3

Verified the given relations.

Used formulae:-

• The standard quadratic equation is ax^2+bx+c = 0

• Sum of the roots (α+ β) = -b/a

• Product of the roots (αβ)=c/a