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

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

Factor

$$-3\left(x-\frac{2-\sqrt{7}}{3}\right)\left(x-\frac{\sqrt{7}+2}{3}\right)$$

Steps Using the Quadratic Formula

Combine 2x and 2x to get 4x.

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

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.

$$-3x^{2}+4x+1=0$$

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.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-3\right)}}{2\left(-3\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-3\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-4±\sqrt{16+12}}{2\left(-3\right)}$$

Add 16 to 12.

$$x=\frac{-4±\sqrt{28}}{2\left(-3\right)}$$

Take the square root of 28.

$$x=\frac{-4±2\sqrt{7}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-4±2\sqrt{7}}{-6}$$

Now solve the equation x=\frac{-4±2\sqrt{7}}{-6} when ± is plus. Add -4 to 2\sqrt{7}.

$$x=\frac{2\sqrt{7}-4}{-6}$$

Divide -4+2\sqrt{7} by -6.

$$x=\frac{2-\sqrt{7}}{3}$$

Now solve the equation x=\frac{-4±2\sqrt{7}}{-6} when ± is minus. Subtract 2\sqrt{7} from -4.

$$x=\frac{-2\sqrt{7}-4}{-6}$$

Divide -4-2\sqrt{7} by -6.

$$x=\frac{\sqrt{7}+2}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2-\sqrt{7}}{3} for x_{1} and \frac{2+\sqrt{7}}{3} for x_{2}.

$$-3x^{2}+4x+1=-3\left(x-\frac{2-\sqrt{7}}{3}\right)\left(x-\frac{\sqrt{7}+2}{3}\right)$$

Evaluate

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

Solution Steps

Combine 2x and 2x to get 4x.

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