Solve x^2-8x+15div3-x

$${ x }^{ 2 } -8x+15 \div 3-x$$

Evaluate

$$x^{2}-9x+5$$

Solution Steps

Divide 15 by 3 to get 5.

$$x^{2}-8x+5-x$$

Combine -8x and -x to get -9x.

$$x^{2}-9x+5$$

Factor

$$\left(x-\frac{9-\sqrt{61}}{2}\right)\left(x-\frac{\sqrt{61}+9}{2}\right)$$

Steps Using the Quadratic Formula

Divide 15 by 3 to get 5.

$$factor(x^{2}-8x+5-x)$$

Combine -8x and -x to get -9x.

$$factor(x^{2}-9x+5)$$

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.

$$x^{2}-9x+5=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 5}}{2}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 5}}{2}$$

Multiply -4 times 5.

$$x=\frac{-\left(-9\right)±\sqrt{81-20}}{2}$$

Add 81 to -20.

$$x=\frac{-\left(-9\right)±\sqrt{61}}{2}$$

The opposite of -9 is 9.

$$x=\frac{9±\sqrt{61}}{2}$$

Now solve the equation x=\frac{9±\sqrt{61}}{2} when ± is plus. Add 9 to \sqrt{61}.

$$x=\frac{\sqrt{61}+9}{2}$$

Now solve the equation x=\frac{9±\sqrt{61}}{2} when ± is minus. Subtract \sqrt{61} from 9.

$$x=\frac{9-\sqrt{61}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9+\sqrt{61}}{2} for x_{1} and \frac{9-\sqrt{61}}{2} for x_{2}.

$$x^{2}-9x+5=\left(x-\frac{\sqrt{61}+9}{2}\right)\left(x-\frac{9-\sqrt{61}}{2}\right)$$