### Video Transcript

Find the coordinates of the vertex of the function π of π₯ equals negative seven π₯ squared plus seven π₯ plus five.

Whenever weβre looking to find the location of the vertex of a quadratic function, we should be thinking about representing our equation in completed square form. Thatβs usually of the form π₯ plus π all squared plus π. And we recall that a function of this form must have a vertex at negative π, π. In fact, we can multiply everything in this set of parentheses by some constant value π and the vertex itself doesnβt change. So how do we write the equation negative seven π₯ squared plus seven π₯ plus five in completed square form?

Well, weβll begin by factoring negative seven from the first two terms. So the first two terms become negative seven times π₯ squared minus π₯. Initially, weβre just going to focus on completing the square for the expression π₯ squared minus π₯. We recall that to do that, we halve the coefficient of π₯.

Now, the coefficient of π₯ here is technically negative one. So halving that, we get negative one-half. We then subtract this value squared. Well, negative a half squared is a quarter. So weβre going to subtract a quarter from this bracket. We replace π₯ squared minus π₯ with this expression in our original function. And we find that π of π₯ is equal to negative seven times π₯ minus a half all squared minus a quarter all plus five.

Weβre now going to redistribute these parentheses. Negative seven multiplied by π₯ minus a half all squared is negative seven π₯ minus a half all squared. We then multiply negative seven by negative a quarter. We get seven-quarters. And, of course, we have this plus five to deal with. This is almost in completed square form. Our final step is going to be to find the sum of seven-quarters and five.

To do that, we write five as five over one. And we create a common denominator by multiplying both the numerator and denominator of this fraction by four. And that gives us seven-quarters plus twenty-quarters, which is of course 27 over four. So our completed square form is negative seven times π₯ minus a half all squared plus 27 over four. We can convert 27 over four into a mixed number by dividing 27 by four to get six and then writing the remainder as a fraction. So 27 over four is the same as six and three-quarters.

To find the coordinates of the vertex, we begin by taking the negative value of negative one-half. So thatβs a half. And then, the π¦-coordinate is simply six and three-quarters. And so, the coordinate weβre looking for is a half, six and three-quarters.