x^2 - 10x + 33 ≡ (x - a)^2 + b. Work out the value of a and b.
x2 - 10x + 33 ≡ (x - a)2 + b Work out the value of a and b. Our aim here is to write the expression on the left in the same form as the one on the right so that we can compare the two. Therefore, consider the expression on the left on its own: x2 - 10x + 33 This is a quadratic expression. We may be able to factorise it into brackets, similar to the expression, (x - a)2, above. If you remember how to factorise a quadratic, the first thing we should try is to find a number which multiplies to give +33 and add to give -10. You will soon discover that there are no whole numbers which will help us here. But remember that the full expression at the top on the right is (x - a)2 + b, therefore if we try subtracting some value b from the +33, we may be able to find two numbers which multiply to give us the remainder and add to give -10. We also know that because the expression (x - a)2 is a perfect square, that the two numbers which add to give -10 will be equal – if you can’t remember why the two numbers will be equal, it shall be demonstrated later. If we divide -10 by 2, then we get -5. If we multiply -5 by -5, we get 25. Therefore, let’s try removing some value from 33 to get 25. x2 - 10x + 33 = x2 - 10x + 25 + 8 Now if we ignore the +8 and focus on the remaining expression, it is now possible to factorise it to get a perfect square, following the steps bellow: = x2 - 5x - 5x + 25 + 8 = x(x - 5) - 5(x - 5) + 8 = (x - 5)(x - 5) + 8 = (x - 5)2 + 8 Therefore, (x - 5)2 + 8 ≡ (x - a)2 + b, and a = 5, b = 8.
