Question

The difference between the roots of the equation x² - 13x + k = 0 is 7 find k.​

Answer 1

$\huge\underline\frak{\fbox{AnSwEr :-}}$

Comparing x² - 13x + k = 0 with ax² + bx + c = 0

a = 1, b = -13 , c = k

Let a and b be the roots of the equation.

$\leadsto$ α + β = $\Large\sf\frac{-b}{a}$ = $\Large\sf\frac{-13}{1}$ = 13 . . . (I)

But α - β = 7 . . . . . . . . . . (given) (II)

$\leadsto$ 2 α = 20 . . . (adding (I) and (II))

$\leadsto$ α = 10

$\leadsto$ 10 + β = 13 . . . (from (I))

$\leadsto$ β = 13 - 10

$\leadsto$ β = 3

But α x β = $\Large\sf\frac{c}{a}$

$\leadsto$ 10 x 3 = $\Large\sf\frac{k}{1}$

$\large\leadsto$ k = 30

Answer 2

$\Huge{\blue{\underline{\textsf{Answer}}}}$

x² - 13x + k = 0

$\large{\green{\underline{\tt{Consider\:a\:and\:b\:are\:the\:roots}}}}$

$\implies\rm x^2+(a+b)x+ab=0$

$\implies\rm Here, a+b=-13 ......(1)$

$\implies\rm a-b=7.....(2)$

$\implies\rm ab=k$

$\large{\red{\underline{\tt{Solve\:eq\:(1)\:and\:(2)}}}}$

$\implies\rm 2a=-6$

$\implies\rm -3$

$\implies\rm b=-10$

∴ ab = -3 × -10 = 30 = k

$\fbox{k=30}$