A triangle has sides of length $x-1,x,x+1.$ The largest angle in the triangle is ${135}^{\circ }.$

Find the value of $x,$ giving your answer in the form $p\sqrt{q}+r,$ where $p,q,r\in \mathbb{Z}.$

Although you end up with a quadratic, you don't need the formula because there is no constant term. Just factor out the $x$ and find the roots.

To get your answer in the form required, you need to rationalise the denominator.