For what real value of $v$ is $/frac{-21-/sqrt{301}}{10}$ a root of $5x^2+21x+v$?

v = 7

is the value for which

x = (-21 - √301) / 10

is a solution to the quadratic equation

5x² + 21x + v = 0

Step-by-step explanation:

Given that

x = (-21 - √301) / 10 ... (1)

is a root of the quadratic equation

5x² + 21x + v = 0 ... (2)

We want to find the value of v foe which the equation is true.

Consider the quadratic formula

x = [-b ± √ (b² - 4av) ]/2a ... (3)

Comparing (3) with (2), notice that

b = 21

2a = 10

=> a = 10/2 = 5

and

b² - 4av = 301

=> 21² - 4 (5) v = 301

-20v = 301 - 441

-20v = - 140

v = - 140 / (-20)

v = 7

That is a = 5, b = 21, and v = 7

The equation is then

5x² + 21x + 7 = 0