Solve 97000=2288x^2+5873x+5440

$$97000=2288 { x }^{ 2 } +5873x+5440$$

Solve for x

$$x = \frac{\sqrt{872449249} - 5873}{4576} \approx 5.171383716$$

$$x=\frac{-\sqrt{872449249}-5873}{4576}\approx -7.738254346$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$2288x^{2}+5873x+5440=97000$$

Subtract 97000 from both sides.

$$2288x^{2}+5873x+5440-97000=0$$

Subtract 97000 from 5440 to get -91560.

$$2288x^{2}+5873x-91560=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2288 for a, 5873 for b, and -91560 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5873±\sqrt{5873^{2}-4\times 2288\left(-91560\right)}}{2\times 2288}$$

Square 5873.

$$x=\frac{-5873±\sqrt{34492129-4\times 2288\left(-91560\right)}}{2\times 2288}$$

Multiply -4 times 2288.

$$x=\frac{-5873±\sqrt{34492129-9152\left(-91560\right)}}{2\times 2288}$$

Multiply -9152 times -91560.

$$x=\frac{-5873±\sqrt{34492129+837957120}}{2\times 2288}$$

Add 34492129 to 837957120.

$$x=\frac{-5873±\sqrt{872449249}}{2\times 2288}$$

Multiply 2 times 2288.

$$x=\frac{-5873±\sqrt{872449249}}{4576}$$

Now solve the equation x=\frac{-5873±\sqrt{872449249}}{4576} when ± is plus. Add -5873 to \sqrt{872449249}.

$$x=\frac{\sqrt{872449249}-5873}{4576}$$

Now solve the equation x=\frac{-5873±\sqrt{872449249}}{4576} when ± is minus. Subtract \sqrt{872449249} from -5873.

$$x=\frac{-\sqrt{872449249}-5873}{4576}$$

The equation is now solved.

$$x=\frac{\sqrt{872449249}-5873}{4576} x=\frac{-\sqrt{872449249}-5873}{4576}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$2288x^{2}+5873x+5440=97000$$

Subtract 5440 from both sides.

$$2288x^{2}+5873x=97000-5440$$

Subtract 5440 from 97000 to get 91560.

$$2288x^{2}+5873x=91560$$

Divide both sides by 2288.

$$\frac{2288x^{2}+5873x}{2288}=\frac{91560}{2288}$$

Dividing by 2288 undoes the multiplication by 2288.

$$x^{2}+\frac{5873}{2288}x=\frac{91560}{2288}$$

Reduce the fraction \frac{91560}{2288} to lowest terms by extracting and canceling out 8.

$$x^{2}+\frac{5873}{2288}x=\frac{11445}{286}$$

Divide \frac{5873}{2288}, the coefficient of the x term, by 2 to get \frac{5873}{4576}. Then add the square of \frac{5873}{4576} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5873}{2288}x+\left(\frac{5873}{4576}\right)^{2}=\frac{11445}{286}+\left(\frac{5873}{4576}\right)^{2}$$

Square \frac{5873}{4576} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5873}{2288}x+\frac{34492129}{20939776}=\frac{11445}{286}+\frac{34492129}{20939776}$$

Add \frac{11445}{286} to \frac{34492129}{20939776} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5873}{2288}x+\frac{34492129}{20939776}=\frac{872449249}{20939776}$$

Factor x^{2}+\frac{5873}{2288}x+\frac{34492129}{20939776}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5873}{4576}\right)^{2}=\frac{872449249}{20939776}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5873}{4576}\right)^{2}}=\sqrt{\frac{872449249}{20939776}}$$

Simplify.

$$x+\frac{5873}{4576}=\frac{\sqrt{872449249}}{4576} x+\frac{5873}{4576}=-\frac{\sqrt{872449249}}{4576}$$

Subtract \frac{5873}{4576} from both sides of the equation.

$$x=\frac{\sqrt{872449249}-5873}{4576} x=\frac{-\sqrt{872449249}-5873}{4576}$$