Why is it that when 2x^2+5x+4=0 is solved both by completing the square and the quadratic equation does the answer not come out the same

Question

Completing the square - Vertex is (-5/4, 11/16)

Quadratic Formula (-5/4, 11/8)

 

This is what I got when I used both of the formulas. I am pretty sure that you are supposed to get the same answer no matter the formula.

 

 

The equation is 2x²+5x+4=0

Answer 1

2x² + 5x + 4 = 0

 

Let's work first by completing the square:

First divide all elements by 2 to get x:

x² + (5/2)x + 2 = 0

x² + (5/2)x = -2

Take half of the coefficient of the x term, square it

and add to both sides. (5/4)² = 25/16

 

x² + (5/2)x + 25/16 = -2 + 25/16

(x + 5/4)² = -7/16

Take square root of both sides

x + 5/4 = ±√(-7/16)

x = -5/4 ± (i√7)/4

x = [-5 ± (i√7)]/4

 

Now let's do the problem by using the quadratic equation.

a = 2, b = 5, c = 4

x = [-b ±√(b²-4ac)]/2a

 

x = [-5 ±√((25-4(2)(4))/2(2)

 

x =[ -5 ±√(-7)]/4

 

x = [-5 ± i√7]/4

 

As can be seen... The answer is the same using either method.

 

 

 

 

Answer 2

Completing the square tells you where the vertex is.  The quadratic formula tells you where the x-intercepts are.  They are not the same.

• The vertex is located at (-5/4, 7/8)
• There are no real x-intercepts; both are complex:  x = (-5/4) ± i(1/4)√7