Solve 11+10x-x^2

$$11 + 10 x - x ^ { 2 }$$

Factor

$$-\left(x-11\right)\left(x+1\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-x^{2}+10x+11$$

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+11. To find a and b, set up a system to be solved.

$$a+b=10 ab=-11=-11$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. The only such pair is the system solution.

$$a=11 b=-1$$

Rewrite -x^{2}+10x+11 as \left(-x^{2}+11x\right)+\left(-x+11\right).

$$\left(-x^{2}+11x\right)+\left(-x+11\right)$$

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

$$-x\left(x-11\right)-\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(-x-1\right)$$

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.

$$-x^{2}+10x+11=0$$

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.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-1\right)\times 11}}{2\left(-1\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-1\right)\times 11}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-10±\sqrt{100+4\times 11}}{2\left(-1\right)}$$

Multiply 4 times 11.

$$x=\frac{-10±\sqrt{100+44}}{2\left(-1\right)}$$

Add 100 to 44.

$$x=\frac{-10±\sqrt{144}}{2\left(-1\right)}$$

Take the square root of 144.

$$x=\frac{-10±12}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-10±12}{-2}$$

Now solve the equation x=\frac{-10±12}{-2} when ± is plus. Add -10 to 12.

$$x=\frac{2}{-2}$$

Divide 2 by -2.

$$x=-1$$

Now solve the equation x=\frac{-10±12}{-2} when ± is minus. Subtract 12 from -10.

$$x=-\frac{22}{-2}$$

Divide -22 by -2.

$$x=11$$

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 11 for x_{2}.

$$-x^{2}+10x+11=-\left(x-\left(-1\right)\right)\left(x-11\right)$$

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

$$-x^{2}+10x+11=-\left(x+1\right)\left(x-11\right)$$

Evaluate

$$-\left(x-11\right)\left(x+1\right)$$