Solve {l}{3x-x^2}{+2x-1}

$$\left. \begin{array} { l } { 3 x - x ^ { 2 } } \\ { + 2 x - 1 } \end{array} \right.$$

Factor

$$-\left(x-\frac{5-\sqrt{21}}{2}\right)\left(x-\frac{\sqrt{21}+5}{2}\right)$$

Steps Using the Quadratic Formula

Combine 3x and 2x to get 5x.

$$factor(5x-x^{2}-1)$$

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.

$$-x^{2}+5x-1=0$$

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.

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

Square 5.

$$x=\frac{-5±\sqrt{25-4\left(-1\right)\left(-1\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-5±\sqrt{25+4\left(-1\right)}}{2\left(-1\right)}$$

Multiply 4 times -1.

$$x=\frac{-5±\sqrt{25-4}}{2\left(-1\right)}$$

Add 25 to -4.

$$x=\frac{-5±\sqrt{21}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-5±\sqrt{21}}{-2}$$

Now solve the equation x=\frac{-5±\sqrt{21}}{-2} when ± is plus. Add -5 to \sqrt{21}.

$$x=\frac{\sqrt{21}-5}{-2}$$

Divide -5+\sqrt{21} by -2.

$$x=\frac{5-\sqrt{21}}{2}$$

Now solve the equation x=\frac{-5±\sqrt{21}}{-2} when ± is minus. Subtract \sqrt{21} from -5.

$$x=\frac{-\sqrt{21}-5}{-2}$$

Divide -5-\sqrt{21} by -2.

$$x=\frac{\sqrt{21}+5}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5-\sqrt{21}}{2} for x_{1} and \frac{5+\sqrt{21}}{2} for x_{2}.

$$-x^{2}+5x-1=-\left(x-\frac{5-\sqrt{21}}{2}\right)\left(x-\frac{\sqrt{21}+5}{2}\right)$$

Evaluate

$$-x^{2}+5x-1$$

Solution Steps

Combine 3x and 2x to get 5x.

$$5x-x^{2}-1$$