Question

If the sum of the zeroes of the quadratic polynomial p(x) = kx² + 5x + 7k, is equal to their product find the value of k.​

Answer 1

Step-by-step explanation:

Given :-

The sum of the zeroes of the quadratic polynomial p(x) = kx² + 5x + 7k is equal to their product .

To find :-

The value of k.

Solution :-

Given quadratic polynomial

p(x) = kx² + 5x + 7k

On comparing with the standard quadratic polynomial ax²+bx+c then

a = k

b = 5

c = 7k

We know that

Sum of the zeroes = -b/a

Sum of the zeroes of the given p(x)

=-5/k

Sum of the zeroes = -5/k -----------(1)

and

Product of the zeroes = c/a

Product of the given p(x)

= 7k/k

= 7

Product of the zeroes = 7 ---------(2)

According to the given problem

Sum of the zeroes = Product of the zeroes

=> equation (1) = equation (2)

=> -5/k = 7

=> -5 = 7×k

=> -5/7 = k

Therefore, k = -5/7

Answer:-

The value of k for the polynomial is -5/7

Used formulae:-

→ The standard quadratic polynomial is ax²+bx+c

→ Sum of the zeroes = -b/a

→ Product of the zeroes = c/a

Answer 2

$\rm \bigstar \: Solution :$

$\rm \: The \: given \: quadratic \: polynomial \: is$

$\rm \implies \: p(x) = k \: {x}^{2} + 5 \: x + 7 \: k$

$\rm \bigstar \: Here :$

$\rm a = k , b = 5 \: and \:c = 7 \: k$

$\rm \bigstar \: Consider :$

$\rm \: Assume \: that \: \alpha \: and \: \beta \: are$

$\rm \: zeroes \: of \: quadratic \: polynomial$

$\rm \: p(x) = k \: {x}^{2} + 5 \: x + 7 \: k$

$\rm \bigstar \: So :$

$\rm : \longmapsto \: Sum \: of \: the \: roots$

$\rm \: \alpha + \beta = \dfrac{ - b}{a} = \dfrac{ - 5}{k}$

$\bf \: \: \: \: \: \: \: \: \: \: \: \: And$

$\rm \: : \longmapsto \: Product \: of \: the \: roots :$

$\rm \alpha \beta = \dfrac{c}{a} = \dfrac{7 \: k}{k} = 7$

$\rm \bigstar \: Since :$

$\small \text{Sum of the roots = product of the roots}$

$\rm \implies \: \alpha + \beta = \alpha \beta$

$\rm \implies\dfrac{ - b}{a} = \dfrac{c}{a}$

$\rm \implies \: \dfrac{ - 5}{k} = \dfrac{7 \: k}{k}$

$\rm \implies \: \dfrac{ - 5}{k} = 7$

$\rm \implies \: k = \dfrac{ - 5}{7}$

$\rm \bigstar \: Therefore :$

$\rm \: The \: value \: of \: k$

$\rm: \longmapsto \: k = \dfrac{ - 5}{7}$