Solve -x^2-2x-1

$$- { x }^{ 2 } -2x-1$$

Factor

$$-\left(x+1\right)^{2}$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=-2 ab=-\left(-1\right)=1$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. The only such pair is the system solution.

$$a=-1 b=-1$$

Rewrite -x^{2}-2x-1 as \left(-x^{2}-x\right)+\left(-x-1\right).

$$\left(-x^{2}-x\right)+\left(-x-1\right)$$

Factor out -x in the first and -1 in the second group.

$$-x\left(x+1\right)-\left(x+1\right)$$

Factor out common term x+1 by using distributive property.

$$\left(x+1\right)\left(-x-1\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-x^{2}-2x-1=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-2\right)±\sqrt{4+4\left(-1\right)}}{2\left(-1\right)}$$

Multiply 4 times -1.

$$x=\frac{-\left(-2\right)±\sqrt{4-4}}{2\left(-1\right)}$$

Add 4 to -4.

$$x=\frac{-\left(-2\right)±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

$$x=\frac{-\left(-2\right)±0}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±0}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±0}{-2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1 for x_{1} and -1 for x_{2}.

$$-x^{2}-2x-1=-\left(x-\left(-1\right)\right)\left(x-\left(-1\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

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

Evaluate

$$-\left(x+1\right)^{2}$$