Solve 0=25x^2-210x-9

$$0 = 25 x ^ { 2 } - 210 x - 9$$

Solve for x

$$x=3\sqrt{2}+\frac{21}{5}\approx 8.442640687$$

$$x=\frac{21}{5}-3\sqrt{2}\approx -0.042640687$$

Steps Using the Quadratic Formula

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

$$25x^{2}-210x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, -210 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-210\right)±\sqrt{\left(-210\right)^{2}-4\times 25\left(-9\right)}}{2\times 25}$$

Square -210.

$$x=\frac{-\left(-210\right)±\sqrt{44100-4\times 25\left(-9\right)}}{2\times 25}$$

Multiply -4 times 25.

$$x=\frac{-\left(-210\right)±\sqrt{44100-100\left(-9\right)}}{2\times 25}$$

Multiply -100 times -9.

$$x=\frac{-\left(-210\right)±\sqrt{44100+900}}{2\times 25}$$

Add 44100 to 900.

$$x=\frac{-\left(-210\right)±\sqrt{45000}}{2\times 25}$$

Take the square root of 45000.

$$x=\frac{-\left(-210\right)±150\sqrt{2}}{2\times 25}$$

The opposite of -210 is 210.

$$x=\frac{210±150\sqrt{2}}{2\times 25}$$

Multiply 2 times 25.

$$x=\frac{210±150\sqrt{2}}{50}$$

Now solve the equation x=\frac{210±150\sqrt{2}}{50} when ± is plus. Add 210 to 150\sqrt{2}.

$$x=\frac{150\sqrt{2}+210}{50}$$

Divide 210+150\sqrt{2} by 50.

$$x=3\sqrt{2}+\frac{21}{5}$$

Now solve the equation x=\frac{210±150\sqrt{2}}{50} when ± is minus. Subtract 150\sqrt{2} from 210.

$$x=\frac{210-150\sqrt{2}}{50}$$

Divide 210-150\sqrt{2} by 50.

$$x=\frac{21}{5}-3\sqrt{2}$$

The equation is now solved.

$$x=3\sqrt{2}+\frac{21}{5} x=\frac{21}{5}-3\sqrt{2}$$

Steps for Completing the Square

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

$$25x^{2}-210x-9=0$$

Add 9 to both sides. Anything plus zero gives itself.

$$25x^{2}-210x=9$$

Divide both sides by 25.

$$\frac{25x^{2}-210x}{25}=\frac{9}{25}$$

Dividing by 25 undoes the multiplication by 25.

$$x^{2}+\left(-\frac{210}{25}\right)x=\frac{9}{25}$$

Reduce the fraction \frac{-210}{25} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{42}{5}x=\frac{9}{25}$$

Divide -\frac{42}{5}, the coefficient of the x term, by 2 to get -\frac{21}{5}. Then add the square of -\frac{21}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{42}{5}x+\left(-\frac{21}{5}\right)^{2}=\frac{9}{25}+\left(-\frac{21}{5}\right)^{2}$$

Square -\frac{21}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{42}{5}x+\frac{441}{25}=\frac{9+441}{25}$$

Add \frac{9}{25} to \frac{441}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{42}{5}x+\frac{441}{25}=18$$

Factor x^{2}-\frac{42}{5}x+\frac{441}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{21}{5}\right)^{2}=18$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{21}{5}\right)^{2}}=\sqrt{18}$$

Simplify.

$$x-\frac{21}{5}=3\sqrt{2} x-\frac{21}{5}=-3\sqrt{2}$$

Add \frac{21}{5} to both sides of the equation.

$$x=3\sqrt{2}+\frac{21}{5} x=\frac{21}{5}-3\sqrt{2}$$