Question

Find all complex solutions of X^2-5X -5= 0

Answer 1

x = [5 + 3√5]/2 or x = [5 -3√5]/2

Explanation:

Points to remember

Solution of a quadratic equation ax² + bx + c = 0

x = [-b ± √(b² - 4ac)]/2a

To find the solutions of given equation

It is given x² - 5x - 5 = 0

here a = 1, b = -5 and c = -5

x = [-b ± √(b² - 4ac)]/2a

= [--5 ± √((-5)² - 4*1*-5)]/2*1

= [5 ± √(25 + 20)]/2

= [5 ± √(45)]/2

= [5 ± 3√5]/2

x = [5 + 3√5]/2 or x = [5 -3√5]/2

Answer 2

`x = ( 5 - 3√( 5) )/(2) \: or \: x = ( 5 +3 √( 5) )/(2)`

EXPLANATION

The given equation is

`{x}^(2) - 5x - 5 = 0`

The solution is given by the formula

`x = \frac{ - b \pm \sqrt{ {b}^(2) - 4ac} }{2a}`

where a=1, b=-5, c=-5

We substitute into the formula to get;

`x = \frac{ - - 5 \pm \sqrt{ {( - 5)}^(2) - 4(1)( - 5)} }{2(1)}`

We simplify to get,

`x = ( 5 \pm √( 45) )/(2)`

The solutions are:

`x = ( 5 - 3√( 5) )/(2) \: or \: x = ( 5 +3 √( 5) )/(2)`

The equation has no complex roots.