You can factor a parabola by finding its roots: if
![p(x)=x^2+bx+c](https://tex.z-dn.net/?f=%20p%28x%29%3Dx%5E2%2Bbx%2Bc%20)
has roots
, then you have the following factorization:
![p(x) = (x-x_1)(x-x_2)](https://tex.z-dn.net/?f=%20p%28x%29%20%3D%20%28x-x_1%29%28x-x_2%29%20)
In order to find the roots, you can use the usual formula
![x_{1,2} = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}](https://tex.z-dn.net/?f=%20x_%7B1%2C2%7D%20%3D%20%5Cdfrac%7B-b%5Cpm%5Csqrt%7Bb%5E2-4ac%7D%7D%7B2a%7D%20)
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}](https://tex.z-dn.net/?f=%20x_%7B1%2C2%7D%20%3D%20%5Cdfrac%7B-2%5Cpm%5Csqrt%7B4%2B4%7D%7D%7B2%7D%20%3D%20%5Cdfrac%7B-2%5Cpm%5Csqrt%7B8%7D%7D%7B2%7D%20%3D%20%5Cdfrac%7B-2%5Cpm2%5Csqrt%7B2%7D%7D%7B2%7D%20%3D%201%20%5Cpm%20%5Csqrt%7B2%7D%20)
So, you can factor
![x^2-2x-1 = (x-1-\sqrt{2})(x-1+\sqrt{2})](https://tex.z-dn.net/?f=%20x%5E2-2x-1%20%3D%20%28x-1-%5Csqrt%7B2%7D%29%28x-1%2B%5Csqrt%7B2%7D%29%20)
The same goes for the second parabola.
As for the third exercise, simply plug the values asking
![f(1.5)=-5.25](https://tex.z-dn.net/?f=%20f%281.5%29%3D-5.25%20)
you get
![f(-1.5) = 1.5c-3 = -5.25](https://tex.z-dn.net/?f=%20f%28-1.5%29%20%3D%201.5c-3%20%3D%20-5.25%20)
Add 3 to both sides:
![1.5c = -2.25](https://tex.z-dn.net/?f=%201.5c%20%3D%20-2.25%20)
Divide both sides by 1.5:
![c = 1.5](https://tex.z-dn.net/?f=%20c%20%3D%201.5%20)