real or imaginary solutions

Question

Find the real or imaginary solutions by completing the square. 2x^2+5x-1=0 

The solution set is _. 

please help me with my college algebra hw. thank you so much 

Answer 1

When completing the square  (x ± c)²    in a quadratic equation, we want to manipulate the equation so that one side becomes a perfect square.  c is some constant.

 

2x² + 5x - 1 = 0

 

 

Let's divide both sides of the equation by the coefficient of x².  The coefficient is 2.  This will help set the stage for completing the square.

 

x² + (5 / 2)x - (1 / 2) = 0

 

We want the left side of the equation to be a perfect square.  Divide the coefficient of the x term by 2.  Then, we square the result.

 

(5 / 2) / 2 = (5 / 2)*(1 / 2) = 5 / 4

 

(5 / 4)² = 25 / 16

 

 

The constant term on the left side needs to be  25/16.   So some number added to -1/2 give us 25/16.

 

25/16 - (-1/2) = 33/16

 

 

Now we add 33/16 on both sides of the equation.

 

x² + (5 / 2)x - (1 / 2) + (33 / 16) = 0 + (33 / 16)

 

x² + (5 / 2)x + (25 / 16) = 33 / 16

 

(x + 5/4)(x + 5/4) = 33 / 16

 

(x + 5/4)² = 33 / 16

 

 

Now, we square-root both sides of the equation.

 

x + 5/4 = ±√(33 / 16)

 

x + 5/4 = ±√(33) / 4

 

 

Subtract 5/4 on both sides of the equation.

 

x = (-5 / 4) ± (√(33) / 4)

 

x = (-5 ± √33) / 4

 

 

The plus/minus sign indicates that we will have two solutions.  Since the number under the square-root is positive, these solutions will be REAL.