Solve 9.202x^2-4.104x+30=0

$$9.202 { x }^{ 2 } -4.104x+30=0$$

Solve for x (complex solution)

$$x=\frac{1026+i\times 2\sqrt{16990581}}{4601}\approx 0.222995001+1.791768418i$$

$$x=\frac{-i\times 2\sqrt{16990581}+1026}{4601}\approx 0.222995001-1.791768418i$$

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.

$$9.202x^{2}-4.104x+30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9.202 for a, -4.104 for b, and 30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4.104\right)±\sqrt{\left(-4.104\right)^{2}-4\times 9.202\times 30}}{2\times 9.202}$$

Square -4.104 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-4.104\right)±\sqrt{16.842816-4\times 9.202\times 30}}{2\times 9.202}$$

Multiply -4 times 9.202.

$$x=\frac{-\left(-4.104\right)±\sqrt{16.842816-36.808\times 30}}{2\times 9.202}$$

Multiply -36.808 times 30.

$$x=\frac{-\left(-4.104\right)±\sqrt{16.842816-1104.24}}{2\times 9.202}$$

Add 16.842816 to -1104.24 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-4.104\right)±\sqrt{-1087.397184}}{2\times 9.202}$$

Take the square root of -1087.397184.

$$x=\frac{-\left(-4.104\right)±\frac{\sqrt{16990581}i}{125}}{2\times 9.202}$$

The opposite of -4.104 is 4.104.

$$x=\frac{4.104±\frac{\sqrt{16990581}i}{125}}{2\times 9.202}$$

Multiply 2 times 9.202.

$$x=\frac{4.104±\frac{\sqrt{16990581}i}{125}}{18.404}$$

Now solve the equation x=\frac{4.104±\frac{\sqrt{16990581}i}{125}}{18.404} when ± is plus. Add 4.104 to \frac{i\sqrt{16990581}}{125}.

$$x=\frac{513+\sqrt{16990581}i}{18.404\times 125}$$

Divide \frac{513+i\sqrt{16990581}}{125} by 18.404 by multiplying \frac{513+i\sqrt{16990581}}{125} by the reciprocal of 18.404.

$$x=\frac{1026+2\sqrt{16990581}i}{4601}$$

Now solve the equation x=\frac{4.104±\frac{\sqrt{16990581}i}{125}}{18.404} when ± is minus. Subtract \frac{i\sqrt{16990581}}{125} from 4.104.

$$x=\frac{-\sqrt{16990581}i+513}{18.404\times 125}$$

Divide \frac{513-i\sqrt{16990581}}{125} by 18.404 by multiplying \frac{513-i\sqrt{16990581}}{125} by the reciprocal of 18.404.

$$x=\frac{-2\sqrt{16990581}i+1026}{4601}$$

The equation is now solved.

$$x=\frac{1026+2\sqrt{16990581}i}{4601} x=\frac{-2\sqrt{16990581}i+1026}{4601}$$

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.

$$9.202x^{2}-4.104x+30=0$$

Subtract 30 from both sides of the equation.

$$9.202x^{2}-4.104x+30-30=-30$$

Subtracting 30 from itself leaves 0.

$$9.202x^{2}-4.104x=-30$$

Divide both sides of the equation by 9.202, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{9.202x^{2}-4.104x}{9.202}=-\frac{30}{9.202}$$

Dividing by 9.202 undoes the multiplication by 9.202.

$$x^{2}+\left(-\frac{4.104}{9.202}\right)x=-\frac{30}{9.202}$$

Divide -4.104 by 9.202 by multiplying -4.104 by the reciprocal of 9.202.

$$x^{2}-\frac{2052}{4601}x=-\frac{30}{9.202}$$

Divide -30 by 9.202 by multiplying -30 by the reciprocal of 9.202.

$$x^{2}-\frac{2052}{4601}x=-\frac{15000}{4601}$$

Divide -\frac{2052}{4601}, the coefficient of the x term, by 2 to get -\frac{1026}{4601}. Then add the square of -\frac{1026}{4601} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2052}{4601}x+\left(-\frac{1026}{4601}\right)^{2}=-\frac{15000}{4601}+\left(-\frac{1026}{4601}\right)^{2}$$

Square -\frac{1026}{4601} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2052}{4601}x+\frac{1052676}{21169201}=-\frac{15000}{4601}+\frac{1052676}{21169201}$$

Add -\frac{15000}{4601} to \frac{1052676}{21169201} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2052}{4601}x+\frac{1052676}{21169201}=-\frac{67962324}{21169201}$$

Factor x^{2}-\frac{2052}{4601}x+\frac{1052676}{21169201}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1026}{4601}\right)^{2}=-\frac{67962324}{21169201}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1026}{4601}\right)^{2}}=\sqrt{-\frac{67962324}{21169201}}$$

Simplify.

$$x-\frac{1026}{4601}=\frac{2\sqrt{16990581}i}{4601} x-\frac{1026}{4601}=-\frac{2\sqrt{16990581}i}{4601}$$

Add \frac{1026}{4601} to both sides of the equation.

$$x=\frac{1026+2\sqrt{16990581}i}{4601} x=\frac{-2\sqrt{16990581}i+1026}{4601}$$