Solve 409287x^2+46436x+878568=5454

$$409287 { x }^{ 2 } +46436x+878568 = 5454$$

Solve for x (complex solution)

$$x=\frac{-23218+\sqrt{356815134194}i}{409287}\approx -0.05672792+1.459464994i$$

$$x=\frac{-\sqrt{356815134194}i-23218}{409287}\approx -0.05672792-1.459464994i$$

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.

$$409287x^{2}+46436x+878568=5454$$

Subtract 5454 from both sides of the equation.

$$409287x^{2}+46436x+878568-5454=5454-5454$$

Subtracting 5454 from itself leaves 0.

$$409287x^{2}+46436x+878568-5454=0$$

Subtract 5454 from 878568.

$$409287x^{2}+46436x+873114=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 409287 for a, 46436 for b, and 873114 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-46436±\sqrt{46436^{2}-4\times 409287\times 873114}}{2\times 409287}$$

Square 46436.

$$x=\frac{-46436±\sqrt{2156302096-4\times 409287\times 873114}}{2\times 409287}$$

Multiply -4 times 409287.

$$x=\frac{-46436±\sqrt{2156302096-1637148\times 873114}}{2\times 409287}$$

Multiply -1637148 times 873114.

$$x=\frac{-46436±\sqrt{2156302096-1429416838872}}{2\times 409287}$$

Add 2156302096 to -1429416838872.

$$x=\frac{-46436±\sqrt{-1427260536776}}{2\times 409287}$$

Take the square root of -1427260536776.

$$x=\frac{-46436±2\sqrt{356815134194}i}{2\times 409287}$$

Multiply 2 times 409287.

$$x=\frac{-46436±2\sqrt{356815134194}i}{818574}$$

Now solve the equation x=\frac{-46436±2\sqrt{356815134194}i}{818574} when ± is plus. Add -46436 to 2i\sqrt{356815134194}.

$$x=\frac{-46436+2\sqrt{356815134194}i}{818574}$$

Divide -46436+2i\sqrt{356815134194} by 818574.

$$x=\frac{-23218+\sqrt{356815134194}i}{409287}$$

Now solve the equation x=\frac{-46436±2\sqrt{356815134194}i}{818574} when ± is minus. Subtract 2i\sqrt{356815134194} from -46436.

$$x=\frac{-2\sqrt{356815134194}i-46436}{818574}$$

Divide -46436-2i\sqrt{356815134194} by 818574.

$$x=\frac{-\sqrt{356815134194}i-23218}{409287}$$

The equation is now solved.

$$x=\frac{-23218+\sqrt{356815134194}i}{409287} x=\frac{-\sqrt{356815134194}i-23218}{409287}$$

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.

$$409287x^{2}+46436x+878568=5454$$

Subtract 878568 from both sides of the equation.

$$409287x^{2}+46436x+878568-878568=5454-878568$$

Subtracting 878568 from itself leaves 0.

$$409287x^{2}+46436x=5454-878568$$

Subtract 878568 from 5454.

$$409287x^{2}+46436x=-873114$$

Divide both sides by 409287.

$$\frac{409287x^{2}+46436x}{409287}=-\frac{873114}{409287}$$

Dividing by 409287 undoes the multiplication by 409287.

$$x^{2}+\frac{46436}{409287}x=-\frac{873114}{409287}$$

Reduce the fraction \frac{-873114}{409287} to lowest terms by extracting and canceling out 3.

$$x^{2}+\frac{46436}{409287}x=-\frac{291038}{136429}$$

Divide \frac{46436}{409287}, the coefficient of the x term, by 2 to get \frac{23218}{409287}. Then add the square of \frac{23218}{409287} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{46436}{409287}x+\left(\frac{23218}{409287}\right)^{2}=-\frac{291038}{136429}+\left(\frac{23218}{409287}\right)^{2}$$

Square \frac{23218}{409287} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{46436}{409287}x+\frac{539075524}{167515848369}=-\frac{291038}{136429}+\frac{539075524}{167515848369}$$

Add -\frac{291038}{136429} to \frac{539075524}{167515848369} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{46436}{409287}x+\frac{539075524}{167515848369}=-\frac{356815134194}{167515848369}$$

Factor x^{2}+\frac{46436}{409287}x+\frac{539075524}{167515848369}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{23218}{409287}\right)^{2}=-\frac{356815134194}{167515848369}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{23218}{409287}\right)^{2}}=\sqrt{-\frac{356815134194}{167515848369}}$$

Simplify.

$$x+\frac{23218}{409287}=\frac{\sqrt{356815134194}i}{409287} x+\frac{23218}{409287}=-\frac{\sqrt{356815134194}i}{409287}$$

Subtract \frac{23218}{409287} from both sides of the equation.

$$x=\frac{-23218+\sqrt{356815134194}i}{409287} x=\frac{-\sqrt{356815134194}i-23218}{409287}$$