Solve y-y^2+3y-2

$$y - y ^ { 2 } + 3 y - 2$$

Factor

$$-\left(y-\left(2-\sqrt{2}\right)\right)\left(y-\left(\sqrt{2}+2\right)\right)$$

Steps Using the Quadratic Formula

Combine y and 3y to get 4y.

$$factor(4y-y^{2}-2)$$

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.

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

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.

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

Square 4.

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

Multiply -4 times -1.

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

Multiply 4 times -2.

$$y=\frac{-4±\sqrt{16-8}}{2\left(-1\right)}$$

Add 16 to -8.

$$y=\frac{-4±\sqrt{8}}{2\left(-1\right)}$$

Take the square root of 8.

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

Multiply 2 times -1.

$$y=\frac{-4±2\sqrt{2}}{-2}$$

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

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

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

$$y=2-\sqrt{2}$$

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

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

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

$$y=\sqrt{2}+2$$

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

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

Evaluate

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

Solution Steps

Combine y and 3y to get 4y.

$$4y-y^{2}-2$$