Solve =-10+10x-x^2

$$= - 10 + 10 x - x ^ { 2 }$$

Factor

$$-\left(x-\left(5-\sqrt{15}\right)\right)\left(x-\left(\sqrt{15}+5\right)\right)$$

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.

$$-x^{2}+10x-10=0$$

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.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\left(-1\right)\left(-10\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-10±\sqrt{100+4\left(-10\right)}}{2\left(-1\right)}$$

Multiply 4 times -10.

$$x=\frac{-10±\sqrt{100-40}}{2\left(-1\right)}$$

Add 100 to -40.

$$x=\frac{-10±\sqrt{60}}{2\left(-1\right)}$$

Take the square root of 60.

$$x=\frac{-10±2\sqrt{15}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-10±2\sqrt{15}}{-2}$$

Now solve the equation x=\frac{-10±2\sqrt{15}}{-2} when ± is plus. Add -10 to 2\sqrt{15}.

$$x=\frac{2\sqrt{15}-10}{-2}$$

Divide -10+2\sqrt{15} by -2.

$$x=5-\sqrt{15}$$

Now solve the equation x=\frac{-10±2\sqrt{15}}{-2} when ± is minus. Subtract 2\sqrt{15} from -10.

$$x=\frac{-2\sqrt{15}-10}{-2}$$

Divide -10-2\sqrt{15} by -2.

$$x=\sqrt{15}+5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5-\sqrt{15} for x_{1} and 5+\sqrt{15} for x_{2}.

$$-x^{2}+10x-10=-\left(x-\left(5-\sqrt{15}\right)\right)\left(x-\left(\sqrt{15}+5\right)\right)$$

Evaluate

$$-x^{2}+10x-10$$