Solve 4/5+25x*36=x

$$\frac{ 4 }{ 5+25x } \times 36=x$$

Solve for x

$$x = \frac{\sqrt{577} - 1}{10} \approx 2.30208243$$

$$x=\frac{-\sqrt{577}-1}{10}\approx -2.50208243$$

Steps Using the Quadratic Formula

Variable x cannot be equal to -\frac{1}{5} since division by zero is not defined. Multiply both sides of the equation by 5\left(5x+1\right).

$$4\times 36=x\times 5\left(5x+1\right)$$

Multiply 4 and 36 to get 144.

$$144=x\times 5\left(5x+1\right)$$

Use the distributive property to multiply x\times 5 by 5x+1.

$$144=25x^{2}+x\times 5$$

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

$$25x^{2}+x\times 5=144$$

Subtract 144 from both sides.

$$25x^{2}+x\times 5-144=0$$

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.

$$25x^{2}+5x-144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, 5 for b, and -144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 25\left(-144\right)}}{2\times 25}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 25\left(-144\right)}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-5±\sqrt{25-100\left(-144\right)}}{2\times 25}$$

Multiply -100 times -144.

$$x=\frac{-5±\sqrt{25+14400}}{2\times 25}$$

Add 25 to 14400.

$$x=\frac{-5±\sqrt{14425}}{2\times 25}$$

Take the square root of 14425.

$$x=\frac{-5±5\sqrt{577}}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{-5±5\sqrt{577}}{50}$$

Now solve the equation x=\frac{-5±5\sqrt{577}}{50} when ± is plus. Add -5 to 5\sqrt{577}.

$$x=\frac{5\sqrt{577}-5}{50}$$

Divide -5+5\sqrt{577} by 50.

$$x=\frac{\sqrt{577}-1}{10}$$

Now solve the equation x=\frac{-5±5\sqrt{577}}{50} when ± is minus. Subtract 5\sqrt{577} from -5.

$$x=\frac{-5\sqrt{577}-5}{50}$$

Divide -5-5\sqrt{577} by 50.

$$x=\frac{-\sqrt{577}-1}{10}$$

The equation is now solved.

$$x=\frac{\sqrt{577}-1}{10} x=\frac{-\sqrt{577}-1}{10}$$

Steps for Completing the Square

Variable x cannot be equal to -\frac{1}{5} since division by zero is not defined. Multiply both sides of the equation by 5\left(5x+1\right).

$$4\times 36=x\times 5\left(5x+1\right)$$

Multiply 4 and 36 to get 144.

$$144=x\times 5\left(5x+1\right)$$

Use the distributive property to multiply x\times 5 by 5x+1.

$$144=25x^{2}+x\times 5$$

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

$$25x^{2}+x\times 5=144$$

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.

$$25x^{2}+5x=144$$

Divide both sides by 25.

$$\frac{25x^{2}+5x}{25}=\frac{144}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$x^{2}+\frac{5}{25}x=\frac{144}{25}$$

Reduce the fraction \frac{5}{25} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{1}{5}x=\frac{144}{25}$$

Divide \frac{1}{5}, the coefficient of the x term, by 2 to get \frac{1}{10}. Then add the square of \frac{1}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{5}x+\left(\frac{1}{10}\right)^{2}=\frac{144}{25}+\left(\frac{1}{10}\right)^{2}$$

Square \frac{1}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=\frac{144}{25}+\frac{1}{100}$$

Add \frac{144}{25} to \frac{1}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{5}x+\frac{1}{100}=\frac{577}{100}$$

Factor x^{2}+\frac{1}{5}x+\frac{1}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{10}\right)^{2}=\frac{577}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{10}\right)^{2}}=\sqrt{\frac{577}{100}}$$

Simplify.

$$x+\frac{1}{10}=\frac{\sqrt{577}}{10} x+\frac{1}{10}=-\frac{\sqrt{577}}{10}$$

Subtract \frac{1}{10} from both sides of the equation.

$$x=\frac{\sqrt{577}-1}{10} x=\frac{-\sqrt{577}-1}{10}$$