Solve 1+1+(x+1)+x+1+(x+1)+x+(x*x)+x

$$1+1+(x+1)+x+1+(x+1)+x+(x \times x)+x$$

Evaluate

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

Solution Steps

Multiply x and x to get x^{2}.

$$1+1+x+1+x+1+x+1+x+x^{2}+x$$

Add 1 and 1 to get 2.

$$2+x+1+x+1+x+1+x+x^{2}+x$$

Add 2 and 1 to get 3.

$$3+x+x+1+x+1+x+x^{2}+x$$

Combine x and x to get 2x.

$$3+2x+1+x+1+x+x^{2}+x$$

Add 3 and 1 to get 4.

$$4+2x+x+1+x+x^{2}+x$$

Combine 2x and x to get 3x.

$$4+3x+1+x+x^{2}+x$$

Add 4 and 1 to get 5.

$$5+3x+x+x^{2}+x$$

Combine 3x and x to get 4x.

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

Combine 4x and x to get 5x.

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

Factor

$$\left(x-\frac{-\sqrt{5}-5}{2}\right)\left(x-\frac{\sqrt{5}-5}{2}\right)$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$factor(1+1+x+1+x+1+x+1+x+x^{2}+x)$$

Add 1 and 1 to get 2.

$$factor(2+x+1+x+1+x+1+x+x^{2}+x)$$

Add 2 and 1 to get 3.

$$factor(3+x+x+1+x+1+x+x^{2}+x)$$

Combine x and x to get 2x.

$$factor(3+2x+1+x+1+x+x^{2}+x)$$

Add 3 and 1 to get 4.

$$factor(4+2x+x+1+x+x^{2}+x)$$

Combine 2x and x to get 3x.

$$factor(4+3x+1+x+x^{2}+x)$$

Add 4 and 1 to get 5.

$$factor(5+3x+x+x^{2}+x)$$

Combine 3x and x to get 4x.

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

Combine 4x and x to get 5x.

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

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.

$$x^{2}+5x+5=0$$

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.

$$x=\frac{-5±\sqrt{5^{2}-4\times 5}}{2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 5}}{2}$$

Multiply -4 times 5.

$$x=\frac{-5±\sqrt{25-20}}{2}$$

Add 25 to -20.

$$x=\frac{-5±\sqrt{5}}{2}$$

Now solve the equation x=\frac{-5±\sqrt{5}}{2} when ± is plus. Add -5 to \sqrt{5}.

$$x=\frac{\sqrt{5}-5}{2}$$

Now solve the equation x=\frac{-5±\sqrt{5}}{2} when ± is minus. Subtract \sqrt{5} from -5.

$$x=\frac{-\sqrt{5}-5}{2}$$

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

$$x^{2}+5x+5=\left(x-\frac{\sqrt{5}-5}{2}\right)\left(x-\frac{-\sqrt{5}-5}{2}\right)$$