Question

if alpha and beta are the roots of x square + 5 x minus 1 is equals to zero then find alpha + beta the whole square​

Answer 1

GIVEN :-

• and are the roots of x² + 5x - 1.

TO FIND :-

• (α+β)²

TO KNOW :-

★ Quadratic Formula ,

$\\ \boxed{ \sf \: roots = \dfrac{ - b± \sqrt{ {b}^{2} - 4ac} }{2a} }$

Here ,

• a → Coefficient of x²
• b → Coefficient of x
• c → Constant

SOLUTION :-

Solving the polynomial x² + 5x - 1 by Quadratic Formula.

• a = 1
• b = 5
• c = -1

$\\ \sf \: roots = \dfrac{ - 5± \sqrt{ {( 5)}^{2} - 4(1)( - 1) } }{2(1)} \\ \\ \\ \sf \: roots = \dfrac{ - 5± \sqrt{25 + 4} }{2} \\ \\ \\ \sf \: roots = \frac{ - 5± \sqrt{29} }{2} \\ \\ \\ \boxed{ \sf \: roots = \frac{ - 5 + \sqrt{29} }{2} , \frac{ - 5 - \sqrt{29} }{2}}$

Let α be (-5+√29)/2 and β be (-5-√29)/2.

♦ α+β = (- 5 + √29)/2 + (- 5 - √29)/2

ㅤㅤㅤ= (- 5 + √29 - 5 - √29)/2

ㅤㅤㅤ= -10/2

ㅤㅤㅤ= -5

So , we have ,

♠ α+β = -5

Squaring both sides , we get..

→ (α+β)² = (-5)²

→ (α+β)² = 25

$\\ \star\underbrace{\boxed{ \sf \: ( \alpha + { \beta )}^{2} = 25 }}$