Sure, I will guide you through this. To prove that the given expression, 4x² - 4x + 6, can only have positive values, let's proceed as follows:
1. Start by analysing the expression. Note that the expression is a quadratic equation. Quadratic equations typically have the form ax² + bx + c. Here a = 4, b = -4, and c = 6.
2. Our objective is to rewrite the expression in the form of a perfect square. The expression can be manipulated into the form 4*(x - d)² + e. We want to find the values for 'd' and 'e' that will make our equation equal to the original one.
3. Let's start with 'b', coefficient at x. It's -4, same as -2*2*d in our targeted formula. So we can find that d = 0.5.
4. Now for 'e', constant term. It's 6 in the original expression and 4*d² + e in our targeted formula. Substitute d=0.5 into 4*d² + e and simplify to find e = 5.5.
5. So, we can rewrite the original expression as 4*(x - 0.5)² + 5.5.
Now, let's consider the behavior of the perfect square term, (x - 0.5)². The minimum value this term can have is 0. This would occur when x equals 0.5.
Due to this, the smallest value our entire expression can have is when we substitute x = 0.5 into it. Doing so results in 4*0 + 5.5, which equals 5.5.
So, we see that the smallest value the original expression, 4x² - 4x + 6, can have is 5.5, which is a positive number. As such, all other values will also be positive. This means that the given expression can only have positive values.
Therefore, we have proved that the expression 4x² - 4x + 6 can only yield positive values.