Solve x^2+8x+10=5!

$${ x }^{ 2 } +8x+10=5 !$$

Solve for x

$$x=3\sqrt{14}-4\approx 7.22497216$$

$$x=-3\sqrt{14}-4\approx -15.22497216$$

Steps Using the Quadratic Formula

The factorial of 5 is 120.

$$x^{2}+8x+10=120$$

Subtract 120 from both sides.

$$x^{2}+8x+10-120=0$$

Subtract 120 from 10 to get -110.

$$x^{2}+8x-110=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -110 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-110\right)}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-110\right)}}{2}$$

Multiply -4 times -110.

$$x=\frac{-8±\sqrt{64+440}}{2}$$

Add 64 to 440.

$$x=\frac{-8±\sqrt{504}}{2}$$

Take the square root of 504.

$$x=\frac{-8±6\sqrt{14}}{2}$$

Now solve the equation x=\frac{-8±6\sqrt{14}}{2} when ± is plus. Add -8 to 6\sqrt{14}.

$$x=\frac{6\sqrt{14}-8}{2}$$

Divide -8+6\sqrt{14} by 2.

$$x=3\sqrt{14}-4$$

Now solve the equation x=\frac{-8±6\sqrt{14}}{2} when ± is minus. Subtract 6\sqrt{14} from -8.

$$x=\frac{-6\sqrt{14}-8}{2}$$

Divide -8-6\sqrt{14} by 2.

$$x=-3\sqrt{14}-4$$

The equation is now solved.

$$x=3\sqrt{14}-4 x=-3\sqrt{14}-4$$

Steps for Completing the Square

The factorial of 5 is 120.

$$x^{2}+8x+10=120$$

Subtract 10 from both sides.

$$x^{2}+8x=120-10$$

Subtract 10 from 120 to get 110.

$$x^{2}+8x=110$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=110+4^{2}$$

Square 4.

$$x^{2}+8x+16=110+16$$

Add 110 to 16.

$$x^{2}+8x+16=126$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+4\right)^{2}=126$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{126}$$

Simplify.

$$x+4=3\sqrt{14} x+4=-3\sqrt{14}$$

Subtract 4 from both sides of the equation.

$$x=3\sqrt{14}-4 x=-3\sqrt{14}-4$$