Solve 5c(c-2)-1=0

$$5 c ( c - 2 ) - 1 = 0$$

Solve for c

$$c=\frac{\sqrt{30}}{5}+1\approx 2.095445115$$

$$c=-\frac{\sqrt{30}}{5}+1\approx -0.095445115$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5c by c-2.

$$5c^{2}-10c-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -10 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$c=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 5\left(-1\right)}}{2\times 5}$$

Square -10.

$$c=\frac{-\left(-10\right)±\sqrt{100-4\times 5\left(-1\right)}}{2\times 5}$$

Multiply -4 times 5.

$$c=\frac{-\left(-10\right)±\sqrt{100-20\left(-1\right)}}{2\times 5}$$

Multiply -20 times -1.

$$c=\frac{-\left(-10\right)±\sqrt{100+20}}{2\times 5}$$

Add 100 to 20.

$$c=\frac{-\left(-10\right)±\sqrt{120}}{2\times 5}$$

Take the square root of 120.

$$c=\frac{-\left(-10\right)±2\sqrt{30}}{2\times 5}$$

The opposite of -10 is 10.

$$c=\frac{10±2\sqrt{30}}{2\times 5}$$

Multiply 2 times 5.

$$c=\frac{10±2\sqrt{30}}{10}$$

Now solve the equation c=\frac{10±2\sqrt{30}}{10} when ± is plus. Add 10 to 2\sqrt{30}.

$$c=\frac{2\sqrt{30}+10}{10}$$

Divide 10+2\sqrt{30} by 10.

$$c=\frac{\sqrt{30}}{5}+1$$

Now solve the equation c=\frac{10±2\sqrt{30}}{10} when ± is minus. Subtract 2\sqrt{30} from 10.

$$c=\frac{10-2\sqrt{30}}{10}$$

Divide 10-2\sqrt{30} by 10.

$$c=-\frac{\sqrt{30}}{5}+1$$

The equation is now solved.

$$c=\frac{\sqrt{30}}{5}+1 c=-\frac{\sqrt{30}}{5}+1$$

Steps for Completing the Square

Use the distributive property to multiply 5c by c-2.

$$5c^{2}-10c-1=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$5c^{2}-10c=1$$

Divide both sides by 5.

$$\frac{5c^{2}-10c}{5}=\frac{1}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$c^{2}+\left(-\frac{10}{5}\right)c=\frac{1}{5}$$

Divide -10 by 5.

$$c^{2}-2c=\frac{1}{5}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$c^{2}-2c+1=\frac{1}{5}+1$$

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

$$c^{2}-2c+1=\frac{6}{5}$$

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

$$\left(c-1\right)^{2}=\frac{6}{5}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(c-1\right)^{2}}=\sqrt{\frac{6}{5}}$$

Simplify.

$$c-1=\frac{\sqrt{30}}{5} c-1=-\frac{\sqrt{30}}{5}$$

Add 1 to both sides of the equation.

$$c=\frac{\sqrt{30}}{5}+1 c=-\frac{\sqrt{30}}{5}+1$$