Solve -x^2+2x+3+(-4x+8)

$$- x ^ { 2 } + 2 x + 3 + ( - 4 x + 8 )$$

Evaluate

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

Solution Steps

Combine 2x and -4x to get -2x.

$$-x^{2}-2x+3+8$$

Add 3 and 8 to get 11.

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

Factor

$$-\left(x-\left(-2\sqrt{3}-1\right)\right)\left(x-\left(2\sqrt{3}-1\right)\right)$$

Steps Using the Quadratic Formula

Combine 2x and -4x to get -2x.

$$factor(-x^{2}-2x+3+8)$$

Add 3 and 8 to get 11.

$$factor(-x^{2}-2x+11)$$

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.

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

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.

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

Square -2.

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

Multiply -4 times -1.

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

Multiply 4 times 11.

$$x=\frac{-\left(-2\right)±\sqrt{4+44}}{2\left(-1\right)}$$

Add 4 to 44.

$$x=\frac{-\left(-2\right)±\sqrt{48}}{2\left(-1\right)}$$

Take the square root of 48.

$$x=\frac{-\left(-2\right)±4\sqrt{3}}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$x=\frac{2±4\sqrt{3}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{2±4\sqrt{3}}{-2}$$

Now solve the equation x=\frac{2±4\sqrt{3}}{-2} when ± is plus. Add 2 to 4\sqrt{3}.

$$x=\frac{4\sqrt{3}+2}{-2}$$

Divide 2+4\sqrt{3} by -2.

$$x=-2\sqrt{3}-1$$

Now solve the equation x=\frac{2±4\sqrt{3}}{-2} when ± is minus. Subtract 4\sqrt{3} from 2.

$$x=\frac{2-4\sqrt{3}}{-2}$$

Divide 2-4\sqrt{3} by -2.

$$x=2\sqrt{3}-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -1-2\sqrt{3} for x_{1} and -1+2\sqrt{3} for x_{2}.

$$-x^{2}-2x+11=-\left(x-\left(-2\sqrt{3}-1\right)\right)\left(x-\left(2\sqrt{3}-1\right)\right)$$