Question

How do u factor this

1) x^2−2x−1
2) x^2−3x+1
3) The function f is defined by f(x)=cx−3, where c is a constant. Find c, if the value of f at x=1.5 is equal to −5.25.
PLEASE SHOW WORK

Answer 1

You can factor a parabola by finding its roots: if

`p(x)=x^2+bx+c`

has roots
`x_1,\ x_2`, then you have the following factorization:

`p(x) = (x-x_1)(x-x_2)`

In order to find the roots, you can use the usual formula

`x_(1,2) = (-b\pm√(b^2-4ac))/(2a)`

In the first example, this formula leads to

`x_(1,2) = (-2\pm√(4+4))/(2) = (-2\pm√(8))/(2) = (-2\pm2√(2))/(2) = 1 \pm √(2)`

So, you can factor

`x^2-2x-1 = (x-1-√(2))(x-1+√(2))`

The same goes for the second parabola.

As for the third exercise, simply plug the values asking

`f(1.5)=-5.25`

you get

`f(-1.5) = 1.5c-3 = -5.25`

Add 3 to both sides:

`1.5c = -2.25`

Divide both sides by 1.5:

`c = 1.5`