Video: Finding the Minimum Value of a Quadratic Expression by Completing the Square

Rewrite the expression π‘₯Β² βˆ’ 12π‘₯ + 20 in the form (π‘₯ + 𝑝)Β² + π‘ž. What is the minimum value of the function 𝑓(π‘₯) = π‘₯Β² βˆ’ 12π‘₯ + 20.

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Video Transcript

Rewrite the expression π‘₯ squared minus 12π‘₯ plus 20 in the form π‘₯ plus 𝑝 all squared plus π‘ž. What is the minimum value of the function 𝑓 of π‘₯ equals π‘₯ squared minus 12π‘₯ plus 20?

This question is essentially asking us to write the expression π‘₯ squared minus 12π‘₯ plus 20 in completed square form. Let’s recap how we do this. When the coefficient of π‘₯ squared is one, we simply halve the coefficient of π‘₯ to find the value of 𝑝. The coefficient of π‘₯ here is negative 12. When we halve that, we get negative six. So, the first part of our completed square form is π‘₯ minus six all squared.

Now, let’s briefly have a look at why we did this. π‘₯ minus six all squared is π‘₯ minus six times π‘₯ minus six. When we just read these parentheses, we get π‘₯ squared minus six π‘₯ minus six π‘₯ plus 36, which is π‘₯ squared minus 12π‘₯ plus 36. We can see that we need π‘₯ squared minus 12π‘₯, which we’ve got, but we want plus 20. So, we’re going to take away this 36. That gives us π‘₯ squared minus 12π‘₯.

Now, of course, we want π‘₯ squared minus 12π‘₯ plus 20, so we need to add 20 on. So, we get π‘₯ minus six all squared minus 36 plus 20. We can simplify negative 36 plus 20 and we get negative 16. And actually, we’re finished. π‘₯ squared minus 12π‘₯ plus 20 is π‘₯ minus six all squared minus 16.

The second part of this question asks us to find the minimum value of the function 𝑓 of π‘₯ equals π‘₯ squared minus 12π‘₯ plus 20. Well, we’ve just written that expression in completed square form. So next, we recall that a function of the form π‘₯ plus 𝑝 all squared plus π‘ž has a vertex at negative π‘π‘ž. When the coefficient of π‘₯ squared is positive, this is a minimum point.

So, comparing our equation to the general form. We find that the function 𝑓 of π‘₯ equals π‘₯ minus six all squared minus 16 has a vertex. And therefore, a minimum point at six, minus 16. The minimum value of the function is the output. In other words, it’s the value of 𝑦. And so, the minimum value of the function 𝑓 of π‘₯ equals π‘₯ squared minus 12π‘₯ plus 20 is negative 16.

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