Hey there!
[tex] \displaystyle \boxed{n = - \frac{ \sqrt{95} }{5} \text{ \: or \: }n = \frac{ \sqrt{95} }{5} }[/tex]
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The given equation is a QUADRATIC EQUATION.
▪️How to recognize a quadratic equation?
⇒Such an equation is written in the following form :
ax² + bx + c = 0 where a ≠ 0
To solve this equation , we can use the quadratic formula :
[tex] \normalsize \textsf{$x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$} [/tex]
According to its discriminant (b² - 4ac), the equation can admit :
- 2 real-number solutions if it is positive.
- 1 real-number solution if it is equal to zero.
- No real-number if it is negative.
Step 1 : Replace the letters in the general quadratic equation with their values in the given equation
a = 5
b = 0
c = -19
Step 2 : Determine the sign of the discriminant:
[tex] {b}^{2} - 4ac \\ \\ \Longrightarrow {0}^{2} - 4(5)( - 19) \\ \\ \Longrightarrow0 - (-380) \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \blue{\Longrightarrow380} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: [/tex]
380 is positive : The equation admits two real-number solutions (also called roots)
Step 3 : Determine the roots of the equation :
[tex] n_1 = \frac{ - b - \sqrt{ {b}^{2} - 4ac} }{2ac} \\ \\ \Longrightarrow \: \frac{ - 0 - \sqrt{380} }{2( 5) } \\ \\ \Longrightarrow \frac{ - \sqrt{380} }{10} \: \: \: \: \: \: \\ \\\Longrightarrow \frac{ - 2 \sqrt{95} }{10} \: \: \: \: \\ \\ \Longrightarrow \boxed{ \red{ \frac{ - \sqrt{95} }{5} }} \: \: [/tex]
[tex] n_2= \frac{ - b + \sqrt{ {b}^{2} - 4ac} }{2ac} \\ \\ \Longrightarrow \: \frac{ - 0 + \sqrt{380} }{2( 5) } \\ \\ \Longrightarrow \frac{ \sqrt{380} }{10} \: \: \: \: \: \: \\ \\\Longrightarrow \frac{ 2 \sqrt{95} }{10} \: \: \: \: \\ \\ \Longrightarrow \boxed{ \green{ \frac{ \sqrt{95} }{5} }} \: \: [/tex]
Note :
[tex] \sqrt{ 380 } \iff \sqrt{ 4 \times 95 } \iff \orange{2\sqrt{95} } [/tex]
▪️Learn more about the quadratic formula :
↣https://brainly.com/question/27638369
