Solve 2x^2+x-2023

$$2 { x }^{ 2 } +x-2023$$

Factor

$$2\left(x-\frac{-\sqrt{16185}-1}{4}\right)\left(x-\frac{\sqrt{16185}-1}{4}\right)$$

Steps Using the Quadratic Formula

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.

$$2x^{2}+x-2023=0$$

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.

$$x=\frac{-1±\sqrt{1^{2}-4\times 2\left(-2023\right)}}{2\times 2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 2\left(-2023\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-1±\sqrt{1-8\left(-2023\right)}}{2\times 2}$$

Multiply -8 times -2023.

$$x=\frac{-1±\sqrt{1+16184}}{2\times 2}$$

Add 1 to 16184.

$$x=\frac{-1±\sqrt{16185}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-1±\sqrt{16185}}{4}$$

Now solve the equation x=\frac{-1±\sqrt{16185}}{4} when ± is plus. Add -1 to \sqrt{16185}.

$$x=\frac{\sqrt{16185}-1}{4}$$

Now solve the equation x=\frac{-1±\sqrt{16185}}{4} when ± is minus. Subtract \sqrt{16185} from -1.

$$x=\frac{-\sqrt{16185}-1}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1+\sqrt{16185}}{4} for x_{1} and \frac{-1-\sqrt{16185}}{4} for x_{2}.

$$2x^{2}+x-2023=2\left(x-\frac{\sqrt{16185}-1}{4}\right)\left(x-\frac{-\sqrt{16185}-1}{4}\right)$$

Evaluate

$$2x^{2}+x-2023$$