Solve 12!x+25x^2=26

$$12 ! x+25 { x }^{ 2 } =26$$

Solve for x

$$x=\frac{\sqrt{2294425328025626}}{5}-9580032\approx 0.000000054$$

$$x=-\frac{\sqrt{2294425328025626}}{5}-9580032\approx -19160064.000000052$$

Steps Using the Quadratic Formula

The factorial of 12 is 479001600.

$$479001600x+25x^{2}=26$$

Subtract 26 from both sides.

$$479001600x+25x^{2}-26=0$$

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.

$$25x^{2}[phone]x-26=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, 479001600 for b, and -26 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-479001600±\sqrt{479001600^{2}-4\times 25\left(-26\right)}}{2\times 25}$$

Square 479001600.

$$x=\frac{-479001600±\sqrt{229442532802560000-4\times 25\left(-26\right)}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-479001600±\sqrt{229442532802560000-100\left(-26\right)}}{2\times 25}$$

Multiply -100 times -26.

$$x=\frac{-479001600±\sqrt{229442532802560000+2600}}{2\times 25}$$

Add 229442532802560000 to 2600.

$$x=\frac{-479001600±\sqrt{229442532802562600}}{2\times 25}$$

Take the square root of 229442532802562600.

$$x=\frac{-479001600±10\sqrt{2294425328025626}}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{-479001600±10\sqrt{2294425328025626}}{50}$$

Now solve the equation x=\frac{-479001600±10\sqrt{2294425328025626}}{50} when ± is plus. Add -479001600 to 10\sqrt{2294425328025626}.

$$x=\frac{10\sqrt{2294425328025626}-479001600}{50}$$

Divide -479001600+10\sqrt{2294425328025626} by 50.

$$x=\frac{\sqrt{2294425328025626}}{5}-9580032$$

Now solve the equation x=\frac{-479001600±10\sqrt{2294425328025626}}{50} when ± is minus. Subtract 10\sqrt{2294425328025626} from -479001600.

$$x=\frac{-10\sqrt{2294425328025626}-479001600}{50}$$

Divide -479001600-10\sqrt{2294425328025626} by 50.

$$x=-\frac{\sqrt{2294425328025626}}{5}-9580032$$

The equation is now solved.

$$x=\frac{\sqrt{2294425328025626}}{5}-9580032 x=-\frac{\sqrt{2294425328025626}}{5}-9580032$$

Steps for Completing the Square

The factorial of 12 is 479001600.

$$479001600x+25x^{2}=26$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$25x^{2}[phone]x=26$$

Divide both sides by 25.

$$\frac{25x^{2}[phone]x}{25}=\frac{26}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$x^{2}+\frac{479001600}{25}x=\frac{26}{25}$$

Divide 479001600 by 25.

$$x^{2}[phone]x=\frac{26}{25}$$

Divide 19160064, the coefficient of the x term, by 2 to get 9580032. Then add the square of 9580032 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}[phone]x+9580032^{2}=\frac{26}{25}+9580032^{2}$$

Square 9580032.

$$x^{2}[phone]x+91777013121024=\frac{26}{25}[phone]$$

Add \frac{26}{25} to 91777013121024.

$$x^{2}[phone]x+91777013121024=\frac{2294425328025626}{25}$$

Factor x^{2}[phone]x+91777013121024. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+9580032\right)^{2}=\frac{2294425328025626}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+9580032\right)^{2}}=\sqrt{\frac{2294425328025626}{25}}$$

Simplify.

$$x+9580032=\frac{\sqrt{2294425328025626}}{5} x+9580032=-\frac{\sqrt{2294425328025626}}{5}$$

Subtract 9580032 from both sides of the equation.

$$x=\frac{\sqrt{2294425328025626}}{5}-9580032 x=-\frac{\sqrt{2294425328025626}}{5}-9580032$$