Question

What is the standard form of `f(x)=(2x-3)(x-2)+(4x-5 )^2`?

Answer 1

The polynomial in standard form is `18x^2-47x+31`.

Explanation:

`f(x)=color(red)((2x-3)(x-2))+color(blue)((4x-5)^2)`

`color(white)(f(x))=color(red)(2x^2-4x-3x+6)+color(blue)((4x-5)(4x-5))`

`color(white)(f(x))=color(red)(2x^2-7x+6)+color(blue)(16x^2-20x-20x+25)`

`color(white)(f(x))=color(red)(2x^2-7x+6)+color(blue)(16x^2-40x+25)`

`color(white)(f(x))=color(red)(2x^2)+color(blue)(16x^2)color(red)(-7x)color(blue)(-40x)+color(red)6+color(blue)(25)`

`color(white)(f(x))=color(purple)(18x^2-47x+31)`

This is the equation of the polynomial in standard form. You can verify this by graphing the original equation and this one and seeing that they are the same parabola.

Answer 2

`f(x)=(2x-3)(x-2)+(4x-5)^2=color(blue)(18x^2-47x+31`

This is the standard form for a quadratic equation:

`ax^2+bx+c`.

Explanation:

`f(x)=(2x-3)(x-2)+(4x-5)^2`

First multiply `(2x-3)` by `(x-2)` using the FOIL method.
https://www.mesacc.edu/~scotz47781/mat120/notes/polynomials/foil_method/foil_method.html

`f(x)=2x^2-7x+6+(4x-5)^2`

Expand `(4x-5)^2` using the FOIL method.

`f(x)=2x^2-7x+6+16x^2-40x+25`

Collect like terms.

`f(x)=(2x^2+16x^2)+(-7x-40x)+(6+25)`

Combine like terms.

`f(x)=18x^2-47x+31` is in standard form for a quadratic equation:

`ax^2+bx+c`,

where:

`a=18`, `b=-47`, `c=31`