Question

For what value of k for which one root of the quadratic equation kx2-14x+8=0 is six times the other

Answer 1

$\bf{\underline{\boxed{\huge{\rm{\red{Answer:}}}}}}$

$\bf{\underline{\underline{\sf{\blue{Given\:equation:}}}}}$

• kx² - 14x + 8 = 0

$\bf{\underline{\underline{\sf{\blue{To\:find:}}}}}$

• Value of k for which one the roots for the given quadratic equations is 6 times the other root.

$\bf{\underline{\underline{\sf{\blue{Solution:}}}}}$

Let α and β be the roots of the equation in variable x.

Let α = x

Let β = 6x

On comparing the given quadratic equations with the general form of quadratic equation.

General form :-

• ax² + bx + c = 0

We obtain the following values of a, b and c.

a = k

b = - 14

c = 8

Now following the question using the sums of roots and product of roots.

Sum of roots = α + β = $\bf\large\frac{-b}{a}$

Product of roots = αβ = $\bf\large\frac{c}{a}$

For sums of roots :-

Plus in the values,

=> x + 6x = $\large\frac{-(-14)}{k}$

=> 7x = $\large\frac{14}{k}$

Cross multiplying,

=> 7x × k = 14

=> k = $\large\frac{14}{7x}$

=> k = $\bf\large\frac{2}{x}$ ---> 1

For product of roots :-

Plug in the values,

=> x (6x) = $\bf\large\frac{8}{k}$

=> 6x² = $\bf\large\frac{8}{k}$

Cross multiplying,

=> 6x² × k = 8

=> k = $\large\frac{8}{6x^2}$

=> k = $\large\frac{4}{3x^2}$ ---> 2

Comparing equations 1 and 2,

=> $\large\frac{2}{x}$ = $\large\frac{4}{3x^2}$

Cross multiplying,

=> (3x²) (2) = (4) (x)

=> 6x² = 4x

=> 6x² - 4x = 0

=> 2x ( 3x - 2) = 0

=> x = 0 OR x = $\large\frac{2}{2/3}$

=> x = 0 OR x = 3

•°• k = 3

•°• value of k is 6 times the other root when, k = 3