Question

Question #06799

Answer 1

See below.

Explanation:

`x^2 -8x-=(x+p)^2+q`

What you have is not an equation it's an identity.

This means it is true for all values of the variable In this case `x`

The expression on the right is in the form:

`a(x-h)^2 + k`

This form is used to find the vertex of a parabola, with `a` being the coefficient of `x^2`, `h` being the axis of symmetry and `k` being the minimum/maximum value. We can find the value of `p` and `q` and thus the maximum/minimum value by transforming:

`x^2-8x` into the form `a(x-h)^2+k`

`x^2 -8x` into form `a(x-h)^2+k`

Bracket of terms containing variable.

`(x^2 -8x)`

Factor out the coefficient of `x^2` if this is not 1.

Add the square of half the coefficient of `x` inside the bracket and subtract it outside the bracket.

`(x^2 -8x +(-4)^2) -(-4)^2`

Make `(x^2 -8x +(-4)^2)` into the square of a binomial.

`(x - 4)^2- (-4)^2=> (x-4)^2-16`

We now have the form `a(x-h)^2 +k`

So `(x+p)^2+q`

is `(x-4)^2-16`

Minimum value is `q`

So:
`q= -16`

Minimum value `-16`

Graph:

graph{x^2 -8x [-20, 20, -23.12, 23.14]}