Solve 546x^2-554x+1621=0

$$546 { x }^{ 2 } -554x+1621=0$$

Solve for x (complex solution)

$$x=\frac{277+\sqrt{808337}i}{546}\approx 0.507326007+1.646658675i$$

$$x=\frac{-\sqrt{808337}i+277}{546}\approx 0.507326007-1.646658675i$$

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.

$$546x^{2}-554x+1621=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 546 for a, -554 for b, and 1621 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-554\right)±\sqrt{\left(-554\right)^{2}-4\times 546\times 1621}}{2\times 546}$$

Square -554.

$$x=\frac{-\left(-554\right)±\sqrt{306916-4\times 546\times 1621}}{2\times 546}$$

Multiply -4 times 546.

$$x=\frac{-\left(-554\right)±\sqrt{306916-2184\times 1621}}{2\times 546}$$

Multiply -2184 times 1621.

$$x=\frac{-\left(-554\right)±\sqrt{306916-3540264}}{2\times 546}$$

Add 306916 to -3540264.

$$x=\frac{-\left(-554\right)±\sqrt{-3233348}}{2\times 546}$$

Take the square root of -3233348.

$$x=\frac{-\left(-554\right)±2\sqrt{808337}i}{2\times 546}$$

The opposite of -554 is 554.

$$x=\frac{554±2\sqrt{808337}i}{2\times 546}$$

Multiply 2 times 546.

$$x=\frac{554±2\sqrt{808337}i}{1092}$$

Now solve the equation x=\frac{554±2\sqrt{808337}i}{1092} when ± is plus. Add 554 to 2i\sqrt{808337}.

$$x=\frac{554+2\sqrt{808337}i}{1092}$$

Divide 554+2i\sqrt{808337} by 1092.

$$x=\frac{277+\sqrt{808337}i}{546}$$

Now solve the equation x=\frac{554±2\sqrt{808337}i}{1092} when ± is minus. Subtract 2i\sqrt{808337} from 554.

$$x=\frac{-2\sqrt{808337}i+554}{1092}$$

Divide 554-2i\sqrt{808337} by 1092.

$$x=\frac{-\sqrt{808337}i+277}{546}$$

The equation is now solved.

$$x=\frac{277+\sqrt{808337}i}{546} x=\frac{-\sqrt{808337}i+277}{546}$$

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.

$$546x^{2}-554x+1621=0$$

Subtract 1621 from both sides of the equation.

$$546x^{2}-554x+1621-1621=-1621$$

Subtracting 1621 from itself leaves 0.

$$546x^{2}-554x=-1621$$

Divide both sides by 546.

$$\frac{546x^{2}-554x}{546}=-\frac{1621}{546}$$

Dividing by 546 undoes the multiplication by 546.

$$x^{2}+\left(-\frac{554}{546}\right)x=-\frac{1621}{546}$$

Reduce the fraction \frac{-554}{546} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{277}{273}x=-\frac{1621}{546}$$

Divide -\frac{277}{273}, the coefficient of the x term, by 2 to get -\frac{277}{546}. Then add the square of -\frac{277}{546} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{277}{273}x+\left(-\frac{277}{546}\right)^{2}=-\frac{1621}{546}+\left(-\frac{277}{546}\right)^{2}$$

Square -\frac{277}{546} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{277}{273}x+\frac{76729}{298116}=-\frac{1621}{546}+\frac{76729}{298116}$$

Add -\frac{1621}{546} to \frac{76729}{298116} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{277}{273}x+\frac{76729}{298116}=-\frac{808337}{298116}$$

Factor x^{2}-\frac{277}{273}x+\frac{76729}{298116}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{277}{546}\right)^{2}=-\frac{808337}{298116}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{277}{546}\right)^{2}}=\sqrt{-\frac{808337}{298116}}$$

Simplify.

$$x-\frac{277}{546}=\frac{\sqrt{808337}i}{546} x-\frac{277}{546}=-\frac{\sqrt{808337}i}{546}$$

Add \frac{277}{546} to both sides of the equation.

$$x=\frac{277+\sqrt{808337}i}{546} x=\frac{-\sqrt{808337}i+277}{546}$$