Solve x+x^2+7-15x

$$x + x ^ { 2 } + 7 - 15 x$$

Factor

$$\left(x-\left(7-\sqrt{42}\right)\right)\left(x-\left(\sqrt{42}+7\right)\right)$$

Steps Using the Quadratic Formula

Combine x and -15x to get -14x.

$$factor(-14x+x^{2}+7)$$

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.

$$x^{2}-14x+7=0$$

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.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 7}}{2}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 7}}{2}$$

Multiply -4 times 7.

$$x=\frac{-\left(-14\right)±\sqrt{196-28}}{2}$$

Add 196 to -28.

$$x=\frac{-\left(-14\right)±\sqrt{168}}{2}$$

Take the square root of 168.

$$x=\frac{-\left(-14\right)±2\sqrt{42}}{2}$$

The opposite of -14 is 14.

$$x=\frac{14±2\sqrt{42}}{2}$$

Now solve the equation x=\frac{14±2\sqrt{42}}{2} when ± is plus. Add 14 to 2\sqrt{42}.

$$x=\frac{2\sqrt{42}+14}{2}$$

Divide 14+2\sqrt{42} by 2.

$$x=\sqrt{42}+7$$

Now solve the equation x=\frac{14±2\sqrt{42}}{2} when ± is minus. Subtract 2\sqrt{42} from 14.

$$x=\frac{14-2\sqrt{42}}{2}$$

Divide 14-2\sqrt{42} by 2.

$$x=7-\sqrt{42}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 7+\sqrt{42} for x_{1} and 7-\sqrt{42} for x_{2}.

$$x^{2}-14x+7=\left(x-\left(\sqrt{42}+7\right)\right)\left(x-\left(7-\sqrt{42}\right)\right)$$

Evaluate

$$x^{2}-14x+7$$

Solution Steps

Combine x and -15x to get -14x.

$$-14x+x^{2}+7$$