Solve x+40%x*90%x=390

$$x+40 \% x \times 90 \% x=390$$

Solve for x

$$x = \frac{5 \sqrt{14065} - 25}{18} \approx 31.554431309$$

$$x=\frac{-5\sqrt{14065}-25}{18}\approx -34.332209087$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$x+\frac{40}{100}x^{2}\times \frac{90}{100}=390$$

Reduce the fraction \frac{40}{100} to lowest terms by extracting and canceling out 20.

$$x+\frac{2}{5}x^{2}\times \frac{90}{100}=390$$

Reduce the fraction \frac{90}{100} to lowest terms by extracting and canceling out 10.

$$x+\frac{2}{5}x^{2}\times \frac{9}{10}=390$$

Multiply \frac{2}{5} times \frac{9}{10} by multiplying numerator times numerator and denominator times denominator.

$$x+\frac{2\times 9}{5\times 10}x^{2}=390$$

Do the multiplications in the fraction \frac{2\times 9}{5\times 10}.

$$x+\frac{18}{50}x^{2}=390$$

Reduce the fraction \frac{18}{50} to lowest terms by extracting and canceling out 2.

$$x+\frac{9}{25}x^{2}=390$$

Subtract 390 from both sides.

$$x+\frac{9}{25}x^{2}-390=0$$

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.

$$\frac{9}{25}x^{2}+x-390=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{9}{25} for a, 1 for b, and -390 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times \frac{9}{25}\left(-390\right)}}{2\times \frac{9}{25}}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times \frac{9}{25}\left(-390\right)}}{2\times \frac{9}{25}}$$

Multiply -4 times \frac{9}{25}.

$$x=\frac{-1±\sqrt{1-\frac{36}{25}\left(-390\right)}}{2\times \frac{9}{25}}$$

Multiply -\frac{36}{25} times -390.

$$x=\frac{-1±\sqrt{1+\frac{2808}{5}}}{2\times \frac{9}{25}}$$

Add 1 to \frac{2808}{5}.

$$x=\frac{-1±\sqrt{\frac{2813}{5}}}{2\times \frac{9}{25}}$$

Take the square root of \frac{2813}{5}.

$$x=\frac{-1±\frac{\sqrt{14065}}{5}}{2\times \frac{9}{25}}$$

Multiply 2 times \frac{9}{25}.

$$x=\frac{-1±\frac{\sqrt{14065}}{5}}{\frac{18}{25}}$$

Now solve the equation x=\frac{-1±\frac{\sqrt{14065}}{5}}{\frac{18}{25}} when ± is plus. Add -1 to \frac{\sqrt{14065}}{5}.

$$x=\frac{\frac{\sqrt{14065}}{5}-1}{\frac{18}{25}}$$

Divide -1+\frac{\sqrt{14065}}{5} by \frac{18}{25} by multiplying -1+\frac{\sqrt{14065}}{5} by the reciprocal of \frac{18}{25}.

$$x=\frac{5\sqrt{14065}-25}{18}$$

Now solve the equation x=\frac{-1±\frac{\sqrt{14065}}{5}}{\frac{18}{25}} when ± is minus. Subtract \frac{\sqrt{14065}}{5} from -1.

$$x=\frac{-\frac{\sqrt{14065}}{5}-1}{\frac{18}{25}}$$

Divide -1-\frac{\sqrt{14065}}{5} by \frac{18}{25} by multiplying -1-\frac{\sqrt{14065}}{5} by the reciprocal of \frac{18}{25}.

$$x=\frac{-5\sqrt{14065}-25}{18}$$

The equation is now solved.

$$x=\frac{5\sqrt{14065}-25}{18} x=\frac{-5\sqrt{14065}-25}{18}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$x+\frac{40}{100}x^{2}\times \frac{90}{100}=390$$

Reduce the fraction \frac{40}{100} to lowest terms by extracting and canceling out 20.

$$x+\frac{2}{5}x^{2}\times \frac{90}{100}=390$$

Reduce the fraction \frac{90}{100} to lowest terms by extracting and canceling out 10.

$$x+\frac{2}{5}x^{2}\times \frac{9}{10}=390$$

Multiply \frac{2}{5} times \frac{9}{10} by multiplying numerator times numerator and denominator times denominator.

$$x+\frac{2\times 9}{5\times 10}x^{2}=390$$

Do the multiplications in the fraction \frac{2\times 9}{5\times 10}.

$$x+\frac{18}{50}x^{2}=390$$

Reduce the fraction \frac{18}{50} to lowest terms by extracting and canceling out 2.

$$x+\frac{9}{25}x^{2}=390$$

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.

$$\frac{9}{25}x^{2}+x=390$$

Divide both sides of the equation by \frac{9}{25}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{9}{25}x^{2}+x}{\frac{9}{25}}=\frac{390}{\frac{9}{25}}$$

Dividing by \frac{9}{25} undoes the multiplication by \frac{9}{25}.

$$x^{2}+\frac{1}{\frac{9}{25}}x=\frac{390}{\frac{9}{25}}$$

Divide 1 by \frac{9}{25} by multiplying 1 by the reciprocal of \frac{9}{25}.

$$x^{2}+\frac{25}{9}x=\frac{390}{\frac{9}{25}}$$

Divide 390 by \frac{9}{25} by multiplying 390 by the reciprocal of \frac{9}{25}.

$$x^{2}+\frac{25}{9}x=\frac{3250}{3}$$

Divide \frac{25}{9}, the coefficient of the x term, by 2 to get \frac{25}{18}. Then add the square of \frac{25}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{25}{9}x+\left(\frac{25}{18}\right)^{2}=\frac{3250}{3}+\left(\frac{25}{18}\right)^{2}$$

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

$$x^{2}+\frac{25}{9}x+\frac{625}{324}=\frac{3250}{3}+\frac{625}{324}$$

Add \frac{3250}{3} to \frac{625}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{25}{9}x+\frac{625}{324}=\frac{351625}{324}$$

Factor x^{2}+\frac{25}{9}x+\frac{625}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{25}{18}\right)^{2}=\frac{351625}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{25}{18}\right)^{2}}=\sqrt{\frac{351625}{324}}$$

Simplify.

$$x+\frac{25}{18}=\frac{5\sqrt{14065}}{18} x+\frac{25}{18}=-\frac{5\sqrt{14065}}{18}$$

Subtract \frac{25}{18} from both sides of the equation.

$$x=\frac{5\sqrt{14065}-25}{18} x=\frac{-5\sqrt{14065}-25}{18}$$