Solve 5x+3+1-x^2-x=

$$5x+3+1- { x }^{ 2 } -x=$$

Evaluate

$$4+4x-x^{2}$$

Solution Steps

Add 3 and 1 to get 4.

$$5x+4-x^{2}-x$$

Combine 5x and -x to get 4x.

$$4x+4-x^{2}$$

Factor

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

Steps Using the Quadratic Formula

Add 3 and 1 to get 4.

$$factor(5x+4-x^{2}-x)$$

Combine 5x and -x to get 4x.

$$factor(4x+4-x^{2})$$

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.

$$-x^{2}+4x+4=0$$

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.

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

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\times 4}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4±\sqrt{16+4\times 4}}{2\left(-1\right)}$$

Multiply 4 times 4.

$$x=\frac{-4±\sqrt{16+16}}{2\left(-1\right)}$$

Add 16 to 16.

$$x=\frac{-4±\sqrt{32}}{2\left(-1\right)}$$

Take the square root of 32.

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

Multiply 2 times -1.

$$x=\frac{-4±4\sqrt{2}}{-2}$$

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

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

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

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

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

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

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

$$x=2\sqrt{2}+2$$

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

$$-x^{2}+4x+4=-\left(x-\left(2-2\sqrt{2}\right)\right)\left(x-\left(2\sqrt{2}+2\right)\right)$$