Given that x^2+10x+3 can be written in the form (x+a)^2+b, find the values of a and b.

We can tackle this question by completing the square. Completing the square allows us to write a quadratic equation (x^2+10x+3) in the simpler form (x+a)^2+b.To complete the square, first we need to find the value of our a. To find the value of our a we need to halve the x term coefficient (halve the number in front of the x). In this question, the number in front of our x is 10 so our a value must be 5.We can then substitute this value of a into our completing the square equation: (x+5)^2+b.To find our b value we need to expand out (x+5)^2 + b and compare this to our original equation.If we expand (x+5)^2 + b we get x^2 + 10x + 25 + b. However, looking at our original equation we know our x^2 and x terms are correct but the constant (the term with no x's) we should be left with is 3 but instead we are left with 25 + b. Hence, 25 + b = 3 (as these are the only terms without x's) so b = -22.Thus, we can write the equation x^2+10x+3 as (x+5)^2 -22 so a = 5 and b = - 22.