Solve x^2+4x+2+x^2-4

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

Evaluate

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

Solution Steps

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

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

Subtract 4 from 2 to get -2.

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

Factor

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

Steps Using the Quadratic Formula

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

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

Subtract 4 from 2 to get -2.

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

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.

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

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.

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

Square 4.

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

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\left(-2\right)}}{2\times 2}$$

Multiply -8 times -2.

$$x=\frac{-4±\sqrt{16+16}}{2\times 2}$$

Add 16 to 16.

$$x=\frac{-4±\sqrt{32}}{2\times 2}$$

Take the square root of 32.

$$x=\frac{-4±4\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±4\sqrt{2}}{4}$$

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

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

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

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

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

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

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

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

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

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