Solve 10x^2+x-9!=0

$$10 { x }^{ 2 } +x-9 ! =0$$

Solve for x

$$x = \frac{\sqrt{14515201} - 1}{20} \approx 190.444100959$$

$$x=\frac{-\sqrt{14515201}-1}{20}\approx -190.544100959$$

Steps Using the Quadratic Formula

The factorial of 9 is 362880.

$$10x^{2}+x-362880=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 1 for b, and -362880 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 10\left(-362880\right)}}{2\times 10}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 10\left(-362880\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-1±\sqrt{1-40\left(-362880\right)}}{2\times 10}$$

Multiply -40 times -362880.

$$x=\frac{-1±\sqrt{1+14515200}}{2\times 10}$$

Add 1 to 14515200.

$$x=\frac{-1±\sqrt{14515201}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-1±\sqrt{14515201}}{20}$$

Now solve the equation x=\frac{-1±\sqrt{14515201}}{20} when ± is plus. Add -1 to \sqrt{14515201}.

$$x=\frac{\sqrt{14515201}-1}{20}$$

Now solve the equation x=\frac{-1±\sqrt{14515201}}{20} when ± is minus. Subtract \sqrt{14515201} from -1.

$$x=\frac{-\sqrt{14515201}-1}{20}$$

The equation is now solved.

$$x=\frac{\sqrt{14515201}-1}{20} x=\frac{-\sqrt{14515201}-1}{20}$$

Steps for Completing the Square

The factorial of 9 is 362880.

$$10x^{2}+x-362880=0$$

Add 362880 to both sides. Anything plus zero gives itself.

$$10x^{2}+x=362880$$

Divide both sides by 10.

$$\frac{10x^{2}+x}{10}=\frac{362880}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\frac{1}{10}x=\frac{362880}{10}$$

Divide 362880 by 10.

$$x^{2}+\frac{1}{10}x=36288$$

Divide \frac{1}{10}, the coefficient of the x term, by 2 to get \frac{1}{20}. Then add the square of \frac{1}{20} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{10}x+\left(\frac{1}{20}\right)^{2}=36288+\left(\frac{1}{20}\right)^{2}$$

Square \frac{1}{20} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{10}x+\frac{1}{400}=36288+\frac{1}{400}$$

Add 36288 to \frac{1}{400}.

$$x^{2}+\frac{1}{10}x+\frac{1}{400}=\frac{14515201}{400}$$

Factor x^{2}+\frac{1}{10}x+\frac{1}{400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{20}\right)^{2}=\frac{14515201}{400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{20}\right)^{2}}=\sqrt{\frac{14515201}{400}}$$

Simplify.

$$x+\frac{1}{20}=\frac{\sqrt{14515201}}{20} x+\frac{1}{20}=-\frac{\sqrt{14515201}}{20}$$

Subtract \frac{1}{20} from both sides of the equation.

$$x=\frac{\sqrt{14515201}-1}{20} x=\frac{-\sqrt{14515201}-1}{20}$$