Prove that two 𝑥 squared plus two 𝑥 plus one is always positive.
To prove a quadratic expression always gives a positive answer, we always complete the square. The reason for this will become apparent in a moment. When we complete the square for a quadratic expression which has one lot of 𝑥 squared, it looks like this: 𝑥 plus 𝑎 all squared plus 𝑏. The coefficient for our 𝑥 squared — that’s the number in front of the 𝑥 squared — is two. So our completed square form is going to look a little bit different.
To complete the square for a more complicated expression like this, we need to begin by forming an expression which has a coefficient of 𝑥 squared of one. To do that, we factorise the whole expression by taking out a factor of two. Two 𝑥 squared divided by two is 𝑥 squared. Two 𝑥 divided by two is 𝑥. And one divided by two is 0.5 or one-half.
Let’s complete the square for the expression inside our brackets. To find the value of the constant inside this first bracket, we halve the coefficient of 𝑥. The coefficient of 𝑥 is one. And a half of one is one-half. We then square this number. One-half squared is a half multiplied by a half, which is a quarter or 0.25.
We need our expression to be plus 0.5. So to achieve this, we’re going to have to add an additional 0.25 into our completed square form. We can say then that, in completed square form, our expression is two lots of 𝑥 plus 0.5 all squared plus 0.25. We can expand this back out, but it isn’t entirely necessary. Two lots of 𝑥 plus 0.5 all squared is just two lots of 𝑥 plus 0.5 all squared. And two lots of 0.25 is 0.5.
Remember, no matter the value, whenever we square a number, it will always be positive. And this is because a positive multiplied by a positive is positive, but also a negative multiplied by a negative is positive. If 𝑥 plus 0.5 all squared is always positive, then two lots of this number must also be always positive. Then we add a positive number. We add 0.5. A positive plus a positive is always positive. And we have shown that our expression two 𝑥 squared plus two 𝑥 plus one is always positive.