Solve x^2-2x+1-5x-5+6

$${ x }^{ 2 } -2x+1-5x-5+6$$

Evaluate

$$x^{2}-7x+2$$

Solution Steps

Combine -2x and -5x to get -7x.

$$x^{2}-7x+1-5+6$$

Subtract 5 from 1 to get -4.

$$x^{2}-7x-4+6$$

Add -4 and 6 to get 2.

$$x^{2}-7x+2$$

Factor

$$\left(x-\frac{7-\sqrt{41}}{2}\right)\left(x-\frac{\sqrt{41}+7}{2}\right)$$

Steps Using the Quadratic Formula

Combine -2x and -5x to get -7x.

$$factor(x^{2}-7x+1-5+6)$$

Subtract 5 from 1 to get -4.

$$factor(x^{2}-7x-4+6)$$

Add -4 and 6 to get 2.

$$factor(x^{2}-7x+2)$$

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.

$$x^{2}-7x+2=0$$

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.

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

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 2}}{2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-7\right)±\sqrt{49-8}}{2}$$

Add 49 to -8.

$$x=\frac{-\left(-7\right)±\sqrt{41}}{2}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{41}}{2}$$

Now solve the equation x=\frac{7±\sqrt{41}}{2} when ± is plus. Add 7 to \sqrt{41}.

$$x=\frac{\sqrt{41}+7}{2}$$

Now solve the equation x=\frac{7±\sqrt{41}}{2} when ± is minus. Subtract \sqrt{41} from 7.

$$x=\frac{7-\sqrt{41}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{41}}{2} for x_{1} and \frac{7-\sqrt{41}}{2} for x_{2}.

$$x^{2}-7x+2=\left(x-\frac{\sqrt{41}+7}{2}\right)\left(x-\frac{7-\sqrt{41}}{2}\right)$$