Solve 90x^2=-10(x^2+0.4x+0.04

$$90 x ^ { 2 } = - 10 ( x ^ { 2 } + 0.4 x + 0.04$$

Solve for x (complex solution)

$$x=-0.02+0.06i$$

$$x=-0.02-0.06i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -10 by x^{2}+0.4x+0.04.

$$90x^{2}=-10x^{2}-4x-0.4$$

Add 10x^{2} to both sides.

$$90x^{2}+10x^{2}=-4x-0.4$$

Combine 90x^{2} and 10x^{2} to get 100x^{2}.

$$100x^{2}=-4x-0.4$$

Add 4x to both sides.

$$100x^{2}+4x=-0.4$$

Add 0.4 to both sides.

$$100x^{2}+4x+0.4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 100 for a, 4 for b, and 0.4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 100\times 0.4}}{2\times 100}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 100\times 0.4}}{2\times 100}$$

Multiply -4 times 100.

$$x=\frac{-4±\sqrt{16-400\times 0.4}}{2\times 100}$$

Multiply -400 times 0.4.

$$x=\frac{-4±\sqrt{16-160}}{2\times 100}$$

Add 16 to -160.

$$x=\frac{-4±\sqrt{-144}}{2\times 100}$$

Take the square root of -144.

$$x=\frac{-4±12i}{2\times 100}$$

Multiply 2 times 100.

$$x=\frac{-4±12i}{200}$$

Now solve the equation x=\frac{-4±12i}{200} when ± is plus. Add -4 to 12i.

$$x=\frac{-4+12i}{200}$$

Divide -4+12i by 200.

$$x=-\frac{1}{50}+\frac{3}{50}i$$

Now solve the equation x=\frac{-4±12i}{200} when ± is minus. Subtract 12i from -4.

$$x=\frac{-4-12i}{200}$$

Divide -4-12i by 200.

$$x=-\frac{1}{50}-\frac{3}{50}i$$

The equation is now solved.

$$x=-\frac{1}{50}+\frac{3}{50}i x=-\frac{1}{50}-\frac{3}{50}i$$

Steps for Completing the Square

Use the distributive property to multiply -10 by x^{2}+0.4x+0.04.

$$90x^{2}=-10x^{2}-4x-0.4$$

Add 10x^{2} to both sides.

$$90x^{2}+10x^{2}=-4x-0.4$$

Combine 90x^{2} and 10x^{2} to get 100x^{2}.

$$100x^{2}=-4x-0.4$$

Add 4x to both sides.

$$100x^{2}+4x=-0.4$$

Divide both sides by 100.

$$\frac{100x^{2}+4x}{100}=-\frac{0.4}{100}$$

Dividing by 100 undoes the multiplication by 100.

$$x^{2}+\frac{4}{100}x=-\frac{0.4}{100}$$

Reduce the fraction \frac{4}{100} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{1}{25}x=-\frac{0.4}{100}$$

Divide -0.4 by 100.

$$x^{2}+\frac{1}{25}x=-0.004$$

Divide \frac{1}{25}, the coefficient of the x term, by 2 to get \frac{1}{50}. Then add the square of \frac{1}{50} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{25}x+\left(\frac{1}{50}\right)^{2}=-0.004+\left(\frac{1}{50}\right)^{2}$$

Square \frac{1}{50} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{25}x+\frac{1}{2500}=-0.004+\frac{1}{2500}$$

Add -0.004 to \frac{1}{2500} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{25}x+\frac{1}{2500}=-\frac{9}{2500}$$

Factor x^{2}+\frac{1}{25}x+\frac{1}{2500}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{50}\right)^{2}=-\frac{9}{2500}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{50}\right)^{2}}=\sqrt{-\frac{9}{2500}}$$

Simplify.

$$x+\frac{1}{50}=\frac{3}{50}i x+\frac{1}{50}=-\frac{3}{50}i$$

Subtract \frac{1}{50} from both sides of the equation.

$$x=-\frac{1}{50}+\frac{3}{50}i x=-\frac{1}{50}-\frac{3}{50}i$$