# Video: AQA GCSE Mathematics Higher Tier Pack 4 β’ Paper 3 β’ Question 27

Prove that 2π₯Β² + 2π₯ + 1 is always positive.

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### Video Transcript

Prove that two π₯ squared plus two π₯ 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 π₯ squared, it looks like this: π₯ plus π all squared plus π. The coefficient for our π₯ squared β thatβs the number in front of the π₯ squared β 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 π₯ squared of one. To do that, we factorise the whole expression by taking out a factor of two. Two π₯ squared divided by two is π₯ squared. Two π₯ divided by two is π₯. And one divided by two is 0.5 or one-half.

Letβ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 π₯. The coefficient of π₯ 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, weβ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 π₯ plus 0.5 all squared plus 0.25. We can expand this back out, but it isnβt entirely necessary. Two lots of π₯ plus 0.5 all squared is just two lots of π₯ 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 π₯ 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 π₯ squared plus two π₯ plus one is always positive.