f(x)= 2 x^{2} +13 x +20

Question

rewrite by completing the square.

Answer 1

2x^2 +13x +20 =

Answer 2

divide by 2 to get

x^2 +13x/2 + 10

take half the x coefficient and square it, then add and subtract it

half 13/2 is 13/4. Square it gives 169/16. Add it to the x^2 and x terms and subtract it from 10

IF you add and subtract the same number the expression remains the same

x^2 +13x/2 + 169/16 + 10- 169/16 = 0

(x-13/4)^2 = 169/16-10 = 9/16

take the square root of both sides

x-13/4 = + or - 3/4

x = 16/4 or 10/4 = 4 or 2.5

Answer 3

f(x) = 2x² + 13x + 20

Step 1: Move the constant term over to the right out of the way:

f(x) = (2x² + 13x ) + 20

Step 2: Factor out whatever is in front of the "x²"

f(x) = 2(x² + (13/2)x ) + 20

Step 3: Add in a number that "completes the square". That number will be the square of 1/2 of the middle coefficient (what's in front of the "x")

In this case the middle coefficient is 13/2. Take 1/2 of it to get 13/4. Then square it to get 169/16. Add in 169/16 HOWEVER, you must also subtract that same amount so that you do not actually change the function. In this case, since the 169/16 is inside a parenthesis and being multiplied by 2, you have actually added (169/16)•2 or 169/8. So you must subtract that same amount like this:

f(x) = 2(x² + (13/2)x + 169/16) + 20 - 169/8

Step 4: Write the quadratic as a "square" and combine the numbers at the end

f(x) = 2(x + 13/4)² + 160/8 - 169/8

f(x) = 2(x + 13/4)² - 9/8

Notice the number grouped with the "x" is 1/2 of the middle coefficient. You can prove this to be truef by multiplying out (x + 13/4)(x + 13/4) to ensure you get x² + (13/2)x + 169/16