X^2+6x-2=0

Question

Quadratic formula

Answer 1

First recall that the quadratic formula used to solve for a quadratic equation of the form   ax² + bx + c = 0   is as follows:

          x = (-b ± √(b² - 4ac))/2a

For the equation  x² + 6x - 2 = 0 ,   a = 1 ,  b = 6 ,  c = -2 . We use these values for a, b, and c in the quadratic formula to solve for x:

     x = (-6 ± √(6² - 4·1·-2))/2·1

       = (-6 ± √(36 + 8))/2

       = (-6 ± √44)/2

Note:     √44 = √4·√11 = 2·√11

     x = (-6 ± √44)/2

       = (-6 ± 2√11)/2

Note:     -6 ± 2√11 = 2·(-3 ± √11)

     x = (-6 ± 2√11)/2

       = 2(-3 ± √11)/2

       = 2(-3 ± √11)/2

       = -3 ± √11

Thus,   x = -3 ± √11

 

Answer 2

The quadratic equation in this problem must be diagnosed with usage of the quadratic formula to arrive at possible solutions:

 

Whereby ax² + bx + c = 0, solutions are (-b + sqrt(b² - 4ac))/2a and (-b - sqrt(b² - 4ac))/2a.

 

In this situation, a = 1, b = 6, and c = -2. By plugging these values into the formula, we arrive at:

 

(-6 + sqrt(36 + 8))/2 and (-6 - sqrt(36 + 8))/2 which correspond to (-6 + 2sqrt(11))/2 and (-6 - 2sqrt(11))/2 when simplifying. Further factoring out of 2 from all terms in each solution gives the following 2 solutions:

 

Solution 1 = -3 + sqrt(11)

Solution 2 = -3 - sqrt(11)

Answer 3

Here is an easier way, in my opinion:

 

x² + 6x - 2 =0

 

x² + 6x = 2

 

Complete the square on the left by finding a number that when added together make 6 and when squared make 9 (3 works); then add that number to both sides:

 

x² + 6x +9 = 2 + 9

 

(x+3)² = 11

 

Take square root of both sides:

 

x+3 = 11^-2

 

Solve for x:

 

x = -3 + √11