Solve 4x+55x-9^2=99999x^2

$$4x+55x- { 9 }^{ 2 } =99999 { x }^{ 2 }$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{32396195}i+59}{199998}\approx 0.000295003-0.028459112i$$

$$x=\frac{59+\sqrt{32396195}i}{199998}\approx 0.000295003+0.028459112i$$

Steps Using the Quadratic Formula

Combine 4x and 55x to get 59x.

$$59x-9^{2}=99999x^{2}$$

Calculate 9 to the power of 2 and get 81.

$$59x-81=99999x^{2}$$

Subtract 99999x^{2} from both sides.

$$59x-81-99999x^{2}=0$$

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.

$$-99999x^{2}+59x-81=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -99999 for a, 59 for b, and -81 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-59±\sqrt{59^{2}-4\left(-99999\right)\left(-81\right)}}{2\left(-99999\right)}$$

Square 59.

$$x=\frac{-59±\sqrt{3481-4\left(-99999\right)\left(-81\right)}}{2\left(-99999\right)}$$

Multiply -4 times -99999.

$$x=\frac{-59±\sqrt{3481+399996\left(-81\right)}}{2\left(-99999\right)}$$

Multiply 399996 times -81.

$$x=\frac{-59±\sqrt{3481-32399676}}{2\left(-99999\right)}$$

Add 3481 to -32399676.

$$x=\frac{-59±\sqrt{-32396195}}{2\left(-99999\right)}$$

Take the square root of -32396195.

$$x=\frac{-59±\sqrt{32396195}i}{2\left(-99999\right)}$$

Multiply 2 times -99999.

$$x=\frac{-59±\sqrt{32396195}i}{-199998}$$

Now solve the equation x=\frac{-59±\sqrt{32396195}i}{-199998} when ± is plus. Add -59 to i\sqrt{32396195}.

$$x=\frac{-59+\sqrt{32396195}i}{-199998}$$

Divide -59+i\sqrt{32396195} by -199998.

$$x=\frac{-\sqrt{32396195}i+59}{199998}$$

Now solve the equation x=\frac{-59±\sqrt{32396195}i}{-199998} when ± is minus. Subtract i\sqrt{32396195} from -59.

$$x=\frac{-\sqrt{32396195}i-59}{-199998}$$

Divide -59-i\sqrt{32396195} by -199998.

$$x=\frac{59+\sqrt{32396195}i}{199998}$$

The equation is now solved.

$$x=\frac{-\sqrt{32396195}i+59}{199998} x=\frac{59+\sqrt{32396195}i}{199998}$$

Steps for Completing the Square

Combine 4x and 55x to get 59x.

$$59x-9^{2}=99999x^{2}$$

Calculate 9 to the power of 2 and get 81.

$$59x-81=99999x^{2}$$

Subtract 99999x^{2} from both sides.

$$59x-81-99999x^{2}=0$$

Add 81 to both sides. Anything plus zero gives itself.

$$59x-99999x^{2}=81$$

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.

$$-99999x^{2}+59x=81$$

Divide both sides by -99999.

$$\frac{-99999x^{2}+59x}{-99999}=\frac{81}{-99999}$$

Dividing by -99999 undoes the multiplication by -99999.

$$x^{2}+\frac{59}{-99999}x=\frac{81}{-99999}$$

Divide 59 by -99999.

$$x^{2}-\frac{59}{99999}x=\frac{81}{-99999}$$

Reduce the fraction \frac{81}{-99999} to lowest terms by extracting and canceling out 9.

$$x^{2}-\frac{59}{99999}x=-\frac{9}{11111}$$

Divide -\frac{59}{99999}, the coefficient of the x term, by 2 to get -\frac{59}{199998}. Then add the square of -\frac{59}{199998} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{59}{99999}x+\left(-\frac{59}{199998}\right)^{2}=-\frac{9}{11111}+\left(-\frac{59}{199998}\right)^{2}$$

Square -\frac{59}{199998} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{59}{99999}x+\frac{3481}{39999200004}=-\frac{9}{11111}+\frac{3481}{39999200004}$$

Add -\frac{9}{11111} to \frac{3481}{39999200004} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{59}{99999}x+\frac{3481}{39999200004}=-\frac{32396195}{39999200004}$$

Factor x^{2}-\frac{59}{99999}x+\frac{3481}{39999200004}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{59}{199998}\right)^{2}=-\frac{32396195}{39999200004}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{59}{199998}\right)^{2}}=\sqrt{-\frac{32396195}{39999200004}}$$

Simplify.

$$x-\frac{59}{199998}=\frac{\sqrt{32396195}i}{199998} x-\frac{59}{199998}=-\frac{\sqrt{32396195}i}{199998}$$

Add \frac{59}{199998} to both sides of the equation.

$$x=\frac{59+\sqrt{32396195}i}{199998} x=\frac{-\sqrt{32396195}i+59}{199998}$$