Write 2x^2 + 16x + 26 in the form a(x + d)^2 + e where a, d, and e are integers.
This is an example of completing the square. Completing the square means taking a usual quadratic equation ax2+bx+c=0 and writing it as a(x+d)2+e=0, where we must find the numbers d and e.Consider completing the square with x2+bx. This is almost a square, because if we multiply out the brackets on the left hand side, we have:(x+b/2)2 =x2+bx +(b/2)2 .This gives us an extra (b/2)2 so if we subtract (b/2)2 from both sides, we have (x+b/2)2-(b/2)2 = x2+bx , and we have completed the square.So, for a quadratic with a =1, we complete the square by halving the coefficient of x to give d, the number inside the brackets. We square this number d and subtract this from the constant c to give the number outside the brackets. Written algebraically this is: x2+bx+c=(x+b/2)2+c-(b/2)2(just adding c to each side in the equation before). Now we have the method, return to our question. Our question is slightly harder than usual. To apply our normal method of completing the square we must take a factor of 2 out. This gives 2[x2+8x+13]. The coefficient of x is 8, so halving this gives 4 and by completing the square on the quadratic within the brackets as above, we have 2[(x+4)2+13-42] =2[(x+4)2-3]. Then expand the outside bracket to give 2[(x+4)2-3]=2(x+4)2-6. This is our answer.A simple way to check is by expanding the brackets in your solution to check if this equals the original quadratic.
