Solve 0=25x^2-36x+41

$$0=25 { x }^{ 2 } -36x+41$$

Solve for x (complex solution)

$$x=\frac{18+\sqrt{701}i}{25}\approx 0.72+1.059056184i$$

$$x=\frac{-\sqrt{701}i+18}{25}\approx 0.72-1.059056184i$$

Steps Using the Quadratic Formula

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

$$25x^{2}-36x+41=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, -36 for b, and 41 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 25\times 41}}{2\times 25}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 25\times 41}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-\left(-36\right)±\sqrt{1296-100\times 41}}{2\times 25}$$

Multiply -100 times 41.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4100}}{2\times 25}$$

Add 1296 to -4100.

$$x=\frac{-\left(-36\right)±\sqrt{-2804}}{2\times 25}$$

Take the square root of -2804.

$$x=\frac{-\left(-36\right)±2\sqrt{701}i}{2\times 25}$$

The opposite of -36 is 36.

$$x=\frac{36±2\sqrt{701}i}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{36±2\sqrt{701}i}{50}$$

Now solve the equation x=\frac{36±2\sqrt{701}i}{50} when ± is plus. Add 36 to 2i\sqrt{701}.

$$x=\frac{36+2\sqrt{701}i}{50}$$

Divide 36+2i\sqrt{701} by 50.

$$x=\frac{18+\sqrt{701}i}{25}$$

Now solve the equation x=\frac{36±2\sqrt{701}i}{50} when ± is minus. Subtract 2i\sqrt{701} from 36.

$$x=\frac{-2\sqrt{701}i+36}{50}$$

Divide 36-2i\sqrt{701} by 50.

$$x=\frac{-\sqrt{701}i+18}{25}$$

The equation is now solved.

$$x=\frac{18+\sqrt{701}i}{25} x=\frac{-\sqrt{701}i+18}{25}$$

Steps for Completing the Square

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

$$25x^{2}-36x+41=0$$

Subtract 41 from both sides. Anything subtracted from zero gives its negation.

$$25x^{2}-36x=-41$$

Divide both sides by 25.

$$\frac{25x^{2}-36x}{25}=-\frac{41}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$x^{2}-\frac{36}{25}x=-\frac{41}{25}$$

Divide -\frac{36}{25}, the coefficient of the x term, by 2 to get -\frac{18}{25}. Then add the square of -\frac{18}{25} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{36}{25}x+\left(-\frac{18}{25}\right)^{2}=-\frac{41}{25}+\left(-\frac{18}{25}\right)^{2}$$

Square -\frac{18}{25} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{36}{25}x+\frac{324}{625}=-\frac{41}{25}+\frac{324}{625}$$

Add -\frac{41}{25} to \frac{324}{625} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{36}{25}x+\frac{324}{625}=-\frac{701}{625}$$

Factor x^{2}-\frac{36}{25}x+\frac{324}{625}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{18}{25}\right)^{2}=-\frac{701}{625}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{18}{25}\right)^{2}}=\sqrt{-\frac{701}{625}}$$

Simplify.

$$x-\frac{18}{25}=\frac{\sqrt{701}i}{25} x-\frac{18}{25}=-\frac{\sqrt{701}i}{25}$$

Add \frac{18}{25} to both sides of the equation.

$$x=\frac{18+\sqrt{701}i}{25} x=\frac{-\sqrt{701}i+18}{25}$$