Solve 5x-10-x^2+10x-5

$$5 x - 10 - x ^ { 2 } + 10 x - 5$$

Evaluate

$$-x^{2}+15x-15$$

Solution Steps

Combine 5x and 10x to get 15x.

$$15x-10-x^{2}-5$$

Subtract 5 from -10 to get -15.

$$15x-15-x^{2}$$

Factor

$$-\left(x-\frac{15-\sqrt{165}}{2}\right)\left(x-\frac{\sqrt{165}+15}{2}\right)$$

Steps Using the Quadratic Formula

Combine 5x and 10x to get 15x.

$$factor(15x-10-x^{2}-5)$$

Subtract 5 from -10 to get -15.

$$factor(15x-15-x^{2})$$

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.

$$-x^{2}+15x-15=0$$

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.

$$x=\frac{-15±\sqrt{15^{2}-4\left(-1\right)\left(-15\right)}}{2\left(-1\right)}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\left(-1\right)\left(-15\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-15±\sqrt{225+4\left(-15\right)}}{2\left(-1\right)}$$

Multiply 4 times -15.

$$x=\frac{-15±\sqrt{225-60}}{2\left(-1\right)}$$

Add 225 to -60.

$$x=\frac{-15±\sqrt{165}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-15±\sqrt{165}}{-2}$$

Now solve the equation x=\frac{-15±\sqrt{165}}{-2} when ± is plus. Add -15 to \sqrt{165}.

$$x=\frac{\sqrt{165}-15}{-2}$$

Divide -15+\sqrt{165} by -2.

$$x=\frac{15-\sqrt{165}}{2}$$

Now solve the equation x=\frac{-15±\sqrt{165}}{-2} when ± is minus. Subtract \sqrt{165} from -15.

$$x=\frac{-\sqrt{165}-15}{-2}$$

Divide -15-\sqrt{165} by -2.

$$x=\frac{\sqrt{165}+15}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{15-\sqrt{165}}{2} for x_{1} and \frac{15+\sqrt{165}}{2} for x_{2}.

$$-x^{2}+15x-15=-\left(x-\frac{15-\sqrt{165}}{2}\right)\left(x-\frac{\sqrt{165}+15}{2}\right)$$