Solve 0=8x^2-200x-1900

$$0 = 8 x ^ { 2 } - 200 x - 1900$$

Solve for x

$$x = \frac{15 \sqrt{7} + 25}{2} \approx 32.343134833$$

$$x=\frac{25-15\sqrt{7}}{2}\approx -7.343134833$$

Steps Using the Quadratic Formula

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

$$8x^{2}-200x-1900=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -200 for b, and -1900 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-200\right)±\sqrt{\left(-200\right)^{2}-4\times 8\left(-1900\right)}}{2\times 8}$$

Square -200.

$$x=\frac{-\left(-200\right)±\sqrt{40000-4\times 8\left(-1900\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-200\right)±\sqrt{40000-32\left(-1900\right)}}{2\times 8}$$

Multiply -32 times -1900.

$$x=\frac{-\left(-200\right)±\sqrt{40000+60800}}{2\times 8}$$

Add 40000 to 60800.

$$x=\frac{-\left(-200\right)±\sqrt{100800}}{2\times 8}$$

Take the square root of 100800.

$$x=\frac{-\left(-200\right)±120\sqrt{7}}{2\times 8}$$

The opposite of -200 is 200.

$$x=\frac{200±120\sqrt{7}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{200±120\sqrt{7}}{16}$$

Now solve the equation x=\frac{200±120\sqrt{7}}{16} when ± is plus. Add 200 to 120\sqrt{7}.

$$x=\frac{120\sqrt{7}+200}{16}$$

Divide 200+120\sqrt{7} by 16.

$$x=\frac{15\sqrt{7}+25}{2}$$

Now solve the equation x=\frac{200±120\sqrt{7}}{16} when ± is minus. Subtract 120\sqrt{7} from 200.

$$x=\frac{200-120\sqrt{7}}{16}$$

Divide 200-120\sqrt{7} by 16.

$$x=\frac{25-15\sqrt{7}}{2}$$

The equation is now solved.

$$x=\frac{15\sqrt{7}+25}{2} x=\frac{25-15\sqrt{7}}{2}$$

Steps for Completing the Square

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

$$8x^{2}-200x-1900=0$$

Add 1900 to both sides. Anything plus zero gives itself.

$$8x^{2}-200x=1900$$

Divide both sides by 8.

$$\frac{8x^{2}-200x}{8}=\frac{1900}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\left(-\frac{200}{8}\right)x=\frac{1900}{8}$$

Divide -200 by 8.

$$x^{2}-25x=\frac{1900}{8}$$

Reduce the fraction \frac{1900}{8} to lowest terms by extracting and canceling out 4.

$$x^{2}-25x=\frac{475}{2}$$

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

$$x^{2}-25x+\left(-\frac{25}{2}\right)^{2}=\frac{475}{2}+\left(-\frac{25}{2}\right)^{2}$$

Square -\frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-25x+\frac{625}{4}=\frac{475}{2}+\frac{625}{4}$$

Add \frac{475}{2} to \frac{625}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-25x+\frac{625}{4}=\frac{1575}{4}$$

Factor x^{2}-25x+\frac{625}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{2}\right)^{2}=\frac{1575}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{2}\right)^{2}}=\sqrt{\frac{1575}{4}}$$

Simplify.

$$x-\frac{25}{2}=\frac{15\sqrt{7}}{2} x-\frac{25}{2}=-\frac{15\sqrt{7}}{2}$$

Add \frac{25}{2} to both sides of the equation.

$$x=\frac{15\sqrt{7}+25}{2} x=\frac{25-15\sqrt{7}}{2}$$