Solve 10800x^2+28561x-28561=0

$$10800 { x }^{ 2 } +28561x-28561=0$$

Solve for x

$$x=\frac{169\sqrt{71761}-28561}{21600}\approx 0.773663511$$

$$x=\frac{-169\sqrt{71761}-28561}{21600}\approx -3.418200548$$

Steps Using the Quadratic Formula

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.

$$10800x^{2}+28561x-28561=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10800 for a, 28561 for b, and -28561 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28561±\sqrt{28561^{2}-4\times 10800\left(-28561\right)}}{2\times 10800}$$

Square 28561.

$$x=\frac{-28561±\sqrt{815730721-4\times 10800\left(-28561\right)}}{2\times 10800}$$

Multiply -4 times 10800.

$$x=\frac{-28561±\sqrt{815730721-43200\left(-28561\right)}}{2\times 10800}$$

Multiply -43200 times -28561.

$$x=\frac{-28561±\sqrt{815730721+1233835200}}{2\times 10800}$$

Add 815730721 to 1233835200.

$$x=\frac{-28561±\sqrt{2049565921}}{2\times 10800}$$

Take the square root of 2049565921.

$$x=\frac{-28561±169\sqrt{71761}}{2\times 10800}$$

Multiply 2 times 10800.

$$x=\frac{-28561±169\sqrt{71761}}{21600}$$

Now solve the equation x=\frac{-28561±169\sqrt{71761}}{21600} when ± is plus. Add -28561 to 169\sqrt{71761}.

$$x=\frac{169\sqrt{71761}-28561}{21600}$$

Now solve the equation x=\frac{-28561±169\sqrt{71761}}{21600} when ± is minus. Subtract 169\sqrt{71761} from -28561.

$$x=\frac{-169\sqrt{71761}-28561}{21600}$$

The equation is now solved.

$$x=\frac{169\sqrt{71761}-28561}{21600} x=\frac{-169\sqrt{71761}-28561}{21600}$$

Steps for Completing the Square

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.

$$10800x^{2}+28561x-28561=0$$

Add 28561 to both sides of the equation.

$$10800x^{2}+28561x-28561-\left(-28561\right)=-\left(-28561\right)$$

Subtracting -28561 from itself leaves 0.

$$10800x^{2}+28561x=-\left(-28561\right)$$

Subtract -28561 from 0.

$$10800x^{2}+28561x=28561$$

Divide both sides by 10800.

$$\frac{10800x^{2}+28561x}{10800}=\frac{28561}{10800}$$

Dividing by 10800 undoes the multiplication by 10800.

$$x^{2}+\frac{28561}{10800}x=\frac{28561}{10800}$$

Divide \frac{28561}{10800}, the coefficient of the x term, by 2 to get \frac{28561}{21600}. Then add the square of \frac{28561}{21600} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{28561}{10800}x+\left(\frac{28561}{21600}\right)^{2}=\frac{28561}{10800}+\left(\frac{28561}{21600}\right)^{2}$$

Square \frac{28561}{21600} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{28561}{10800}x+\frac{815730721}{466560000}=\frac{28561}{10800}+\frac{815730721}{466560000}$$

Add \frac{28561}{10800} to \frac{815730721}{466560000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{28561}{10800}x+\frac{815730721}{466560000}=\frac{2049565921}{466560000}$$

Factor x^{2}+\frac{28561}{10800}x+\frac{815730721}{466560000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{28561}{21600}\right)^{2}=\frac{2049565921}{466560000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{28561}{21600}\right)^{2}}=\sqrt{\frac{2049565921}{466560000}}$$

Simplify.

$$x+\frac{28561}{21600}=\frac{169\sqrt{71761}}{21600} x+\frac{28561}{21600}=-\frac{169\sqrt{71761}}{21600}$$

Subtract \frac{28561}{21600} from both sides of the equation.

$$x=\frac{169\sqrt{71761}-28561}{21600} x=\frac{-169\sqrt{71761}-28561}{21600}$$