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

$x_{1,2} = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

In the first example, this formula leads to

$x_{1,2} = \dfrac{-2\pm\sqrt{4+4}}{2} = \dfrac{-2\pm\sqrt{8}}{2} = \dfrac{-2\pm2\sqrt{2}}{2} = 1 \pm \sqrt{2}$

So, you can factor

$x^2-2x-1 = (x-1-\sqrt{2})(x-1+\sqrt{2})$

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:

$c = 1.5$