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

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

Rewrite the expression π₯Β² β 12π₯ + 20 in the form (π₯ + π)Β² + π. What is the minimum value of the function π(π₯) = π₯Β² β 12π₯ + 20.

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### Video Transcript

Rewrite the expression π₯ squared minus 12π₯ plus 20 in the form π₯ plus π all squared plus π. What is the minimum value of the function π 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. Letβ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 π. 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, letβ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 weβve got, but we want plus 20. So, weβ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, weβ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 π of π₯ equals π₯ squared minus 12π₯ plus 20. Well, weβve just written that expression in completed square form. So next, we recall that a function of the form π₯ plus π all squared plus π has a vertex at negative ππ. 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 π 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, itβs the value of π¦. And so, the minimum value of the function π of π₯ equals π₯ squared minus 12π₯ plus 20 is negative 16.

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