F (x) = 4x>2+12x+9 Rewrite to function by completing the square F (x) = ... (x + ...) >2 + ...

f (x) = 4 (x + 3/2) ^2

Step-by-step explanation:

F (x) = 4 x^2 + 12x + 9

= 4 [ x^2 + 3x] + 9

Add (3/2) ^2 - (3/2) ^2 to the square bracket, we have:

4 [ x^2 + 3x + (3/2) ^2 - (3/2) ^2] + 9

4[ (x + 3/2) ^2 - 9/4] + 9

4 (x + 3/2) ^2 - 9/4 * 4 + 9

4 (x + 3/2) ^2 - 9 + 9

4 (x + 3/2) ^2 + 0

4 (x + 3/2) ^2

Note the (3/2) ^2 - (3/2) ^2 is decoded by dividing b by 2 and squaring it and then subtracting same. Look at the general equation of a circle given below

ax^2 + bx + c

To form a square expression we adopt (b/2) ^2 - (b/2) ^2 to the general equation provided a = 1;

To make a = 1; we had to factor out 4 from the coefficient of x; making perfect square is of concern when we have ax^2 + bx; irrespective of 'c' or not.

This is usually referred to as method of completing the square in solving quadratic equation