Solve x^2+3x+x-1

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

Factor

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

Steps Using the Quadratic Formula

Combine 3x and x to get 4x.

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

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.

$$x^{2}+4x-1=0$$

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.

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

Square 4.

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

Multiply -4 times -1.

$$x=\frac{-4±\sqrt{16+4}}{2}$$

Add 16 to 4.

$$x=\frac{-4±\sqrt{20}}{2}$$

Take the square root of 20.

$$x=\frac{-4±2\sqrt{5}}{2}$$

Now solve the equation x=\frac{-4±2\sqrt{5}}{2} when ± is plus. Add -4 to 2\sqrt{5}.

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

Divide -4+2\sqrt{5} by 2.

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

Now solve the equation x=\frac{-4±2\sqrt{5}}{2} when ± is minus. Subtract 2\sqrt{5} from -4.

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

Divide -4-2\sqrt{5} by 2.

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

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

$$x^{2}+4x-1=\left(x-\left(\sqrt{5}-2\right)\right)\left(x-\left(-\sqrt{5}-2\right)\right)$$

Evaluate

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

Solution Steps

Combine 3x and x to get 4x.

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