Question

If one root of the quadratic equation kx2-14x+8=0 is 6 times the other, then find the value of k.

Answer 1

Solution :

$\bf{\red{\underline{\bf{Given\::}}}}$

If one root of the quadratic equation kx² - 14x + 8 = 0 is 6 times the other.

$\bf{\red{\underline{\bf{To\:find\::}}}}$

The value of k.

$\bf{\red{\underline{\bf{Explanation\::}}}}$

We have two zeroes of the given polynomial α and β.

Let the one zero be 6α

Let the other zero be β

$\leadsto\sf{\beta =6\alpha }$

As the given quadratic polynomial as we compared with ax² + bx + c

• a = 1
• b = -14
• c = 8

So;

$\underline{\pink{\mathcal{SUM\:OF\:THE\:ZEROES\::}}}$

$\mapsto\sf{\alpha +\beta =\dfrac{-b}{a} =\dfrac{Coefficient\:of\:x}{Coefficient\:of\:(x)^{2} } }\\\\\\\mapsto\sf{\alpha +\beta =\dfrac{-(-14)}{k} }\\\\\\\mapsto\sf{\alpha +\beta =\dfrac{14}{k} }\\\\\\\mapsto\sf{\alpha +6\alpha =\dfrac{14}{k} \:\:\:[from(1)]}\\\\\\\mapsto\sf{7\alpha =\dfrac{14}{k} }\\\\\\\mapsto\sf{k\alpha =\cancel{\dfrac{14}{7} }}\\\\\\\mapsto\sf{\orange{\alpha =\dfrac{2}{k} }}$

$\underline{\pink{\mathcal{PRODUCT\:OF\:THE\:ZEROES\::}}}$

$\mapsto\sf{\alpha \beta =\dfrac{c}{a} =\dfrac{Constant\:term}{Coefficient\:of\:(x)^{2} } }\\\\\\\mapsto\sf{\alpha \beta =\dfrac{8}{k} }\\\\\\\mapsto\sf{\alpha (6\alpha )=\dfrac{8}{k} }\\\\\\\mapsto\sf{6\alpha ^{2} =\dfrac{8}{k} }\\\\\\\mapsto\sf{k\alpha ^{2} =\cancel{\dfrac{8}{6} }}\\\\\\\mapsto\sf{k\alpha ^{2} =\dfrac{4}{3} }\\\\\\\mapsto\sf{k\bigg(\dfrac{2}{k}\bigg) ^{2} =\dfrac{4}{3} }\\\\\\\mapsto\sf{\cancel{k}\times \dfrac{4}{\cancel{k}^{2} } =\dfrac{4}{3} }$

$\mapsto\sf{4k=12}\\\\\\\mapsto\sf{k=\cancel{\dfrac{12}{4} }}\\\\\\\mapsto\sf{\orange{k=3}}$

Thus;

The value of k is 3 .

Answer 2

Let roots be α and β

A/q

α = 6β

now, if α and β are roots then equation will be (x -α)(x -β) =0

(x -α)(x -β) =0

⇒ x² - (α+β)x + αβ =0

now putting α = 6β ,

⇒x² - (6β +β)x + 6β×β =0

⇒x² - 7βx +6β² =0

now comparing with kx² -14x +8 =0

7β =14/k

⇒β =2/k

⇒β² = 4/k²_______(1)

and 6β² =8/k

⇒β² =4/3k_______(2)

equating (1) and (2), we get,

4/k² = 4/3k

⇒k =3