Solve -x^2-6x+55

$$- { x }^{ 2 } -6x+55$$

Factor

$$\left(5-x\right)\left(x+11\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+55. To find a and b, set up a system to be solved.

$$a+b=-6 ab=-55=-55$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -55.

$$1,-55 5,-11$$

Calculate the sum for each pair.

$$1-55=-54 5-11=-6$$

The solution is the pair that gives sum -6.

$$a=5 b=-11$$

Rewrite -x^{2}-6x+55 as \left(-x^{2}+5x\right)+\left(-11x+55\right).

$$\left(-x^{2}+5x\right)+\left(-11x+55\right)$$

Factor out x in the first and 11 in the second group.

$$x\left(-x+5\right)+11\left(-x+5\right)$$

Factor out common term -x+5 by using distributive property.

$$\left(-x+5\right)\left(x+11\right)$$

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.

$$-x^{2}-6x+55=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-1\right)\times 55}}{2\left(-1\right)}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-1\right)\times 55}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-6\right)±\sqrt{36+4\times 55}}{2\left(-1\right)}$$

Multiply 4 times 55.

$$x=\frac{-\left(-6\right)±\sqrt{36+220}}{2\left(-1\right)}$$

Add 36 to 220.

$$x=\frac{-\left(-6\right)±\sqrt{256}}{2\left(-1\right)}$$

Take the square root of 256.

$$x=\frac{-\left(-6\right)±16}{2\left(-1\right)}$$

The opposite of -6 is 6.

$$x=\frac{6±16}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{6±16}{-2}$$

Now solve the equation x=\frac{6±16}{-2} when ± is plus. Add 6 to 16.

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

Divide 22 by -2.

$$x=-11$$

Now solve the equation x=\frac{6±16}{-2} when ± is minus. Subtract 16 from 6.

$$x=-\frac{10}{-2}$$

Divide -10 by -2.

$$x=5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -11 for x_{1} and 5 for x_{2}.

$$-x^{2}-6x+55=-\left(x-\left(-11\right)\right)\left(x-5\right)$$

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

$$-x^{2}-6x+55=-\left(x+11\right)\left(x-5\right)$$

Evaluate

$$\left(5-x\right)\left(x+11\right)$$